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GEOMETRY UNIT 3

WORKBOOK

WINTER 2015

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CHAPTER 7 Proportions & Similarity

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Geometry Section 7.1 Notes: Ratios and Proportions A ratio is a comparison of two quantities using ______________________. The ratio of quantities a and b can be expressed as a to b,

a : b, or ab

, where 0b ≠ . Ratios are usually expressed in ________________________________________.

Example 1: a) The number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. b) The country with the longest school year is China, with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.) Extended Ratios can be used to compare ________________________________________ quantities. The expression a : b : c means that the ratio of the first two quantities is a : b, the ratio of the last two quantities is b : c, and the ratio of the first and last quantities is a : c. Example 2: a) In ΔEFG, the ratio of the measures of the angles is 5:12:13. Find the measures of the angles. b) The ratios of the angles in ΔABC is 3:5:7. Find the measure of the angles.

An equation stating that two ratios are equal is called a ________________________________________. In the proportion a cb d=

the numbers a and d are called the extremes of the proportion, ,while the numbers b and c are called the means of the proportion. The product of the extremes ad and the means bc are called the ________________________________________.

The converse of the Cross Products Property is also true. If ad = bc and 0b ≠ and 0d ≠ , then a cb d= . That is, a

band c

dform a

proportion. You can use the Cross Products Property to ________________________________________.

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Example 3: Solve the proportion.

a) 6 918.2 y

= b) 4 5 263 6

x − −=

c) 7 1 15.58 2

n −=

Example 4: a) Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat. b) Brittany randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job.

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Geometry Name: _____________________________________ Section 7.1 Practice Worksheet 1. FOOTBALL A tight end scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game.

2. EDUCATION In a schedule of 6 classes, Marta has 2 elective classes. What is the ratio of elective to non–elective classes in Marta’s schedule?

3. BIOLOGY Out of 274 listed species of birds in the United States, 78 species made the endangered list. Find the ratio of endangered species of birds to listed species in the United States.

4. BOARD GAMES Myra is playing a board game. After 12 turns, Myra has landed on a blue space 3 times. If the game will last for 100 turns, predict how many times Myra will land on a blue space.

5. SCHOOL The ratio of male students to female students in the drama club at Campbell High School is 3:4. If the number of male students in the club is 18, predict the number of female students?

For numbers 6 – 11, solve each proportion.

6. 25 40

x= 7. 7 21

10 x= 8. 20 4

5 6x

=

9. 5 354 8x= 10. 1 7

3 2x +

= 11. 15 33 5

x −=

12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. Find the measures of each side of the triangle.

13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters. What are the measures of the sides of the triangle?

14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet. What are the measures of the sides of the triangle?

15. The ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches. Find the measures of each side of the triangle.

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16. TRIANGLES The ratios of the measures of the angles in △DEF is 7:13:16.

Find the measure of the angles

17. RATIONS Sixteen students went on a week-long hiking trip. They brought with them 320 specially baked, protein-rich, cookies.

What is the ratio of cookies to students? 18. CLOVERS Nathaniel is searching for a four-leaf clover in a field. He finds 2 four-leaf clovers during the first 12 minutes of his

search. If Nathaniel spends a total of 180 minutes searching in the field, predict the number of four-leaf clovers Nathaniel will find. 19. CARS A car company builds two versions of one of its models—a sedan and a station wagon. The ratio of sedans to station

wagons is 11:2. A freighter begins unloading the cars at a dock. Tom counts 18 station wagons and then overhears a dock worker call out, “Okay, that’s all of the wagons . . . bring out the sedans!” How many sedans were on the ship?

20. DISASTER READINESS The town of Oyster Bay

is conducting a survey of 80 households to see how prepared its citizens are for a natural disaster. Of those households surveyed, 66 have a survival kit at home.

a) Write the ratio of people with survival kits in the survey. b) Write the ratio of people without survival kits in the survey. c) There are 29,000 households in Oyster Bay.

If the town wishes to purchase survival kits for all households that do not currently have one, predict the number of kits it will have to purchase

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Geometry Section 7.2 Notes: Similar Polygons Similar Polygons have the same __________________ but not necessarily the same __________________.

As with congruence statements, the order of the vertices in a similarity statement like ABCD ~ WXYZ is important. It identifies the corresponding angles and sides. Example 1: a) If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides. b) If ΔGHK ~ ΔPQR, determine which of the following similarity statements is not true. a) ∠HGK ≅ ∠QPR b) c) ∠K ≅ ∠R d) ∠GHK ≅ ∠QPR The ratio of the lengths of corresponding sides in two similar polygons is called the __________________________________.

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Example 2: a) Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu: b) Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu: Example 3: a) The two polygons are similar. Find the values of x and y. b) The two polygons are similar. Solve for a and b .

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Example 4: If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon. b) If LMNOP ~ VWXYZ, find the perimeter of each polygon.

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Geometry Name: _____________________________________ Section 7.2 Practice Worksheet For numbers 1 and 2, determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning.

1. 2.

For numbers 3 and 4, each pair of polygons is similar. Find the value of x.

3. 4.

5. PENTAGONS If ABCDE ∼ PQRST, find the scale factor of

ABCDE to PQRST and the perimeter of each polygon. 6. SWIMMING POOLS The Minnitte family and the

neighboring Gaudet family both have in-ground swimming pools. The Minnitte family pool, PQRS, measures 48 feet by 84 feet. The Gaudet family pool, WXYZ, measures 40 feet by 70 feet. Are the two pools similar? If so, write the similarity statement and scale factor.

7. PANELS When closed, an entertainment center is made of four square panels. The three smaller panels are congruent squares. What is the scale factor of the larger square to one of the smaller squares?

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8. WIDESCREEN TELEVISIONS An electronics company manufactures widescreen television sets in several different sizes. The rectangular viewing area of each television size is similar to the viewing areas of the other sizes. The company’s 42-inch widescreen television has a viewing area perimeter of approximately 144.4 inches. What is the viewing area perimeter of the company’s 46-inch widescreen television?

9. ICE HOCKEY An official Olympic-sized ice hockey rink measures 30 meters by 60 meters. The ice hockey rink at the local

community college measures 25.5 meters by 51 meters. Are the ice hockey rinks similar? Explain your reasoning. 10. ENLARGING Camille wants to make a pattern for a four-pointed star with dimensions twice as long as the one shown. Help her

by drawing a star with dimensions twice as long on the grid below. 11. BIOLOGY A paramecium is a small single-cell organism. The paramecium magnified below is actually one tenth of a millimeter

long.

a) If you want to make a photograph of the original paramecium so that its image is 1 centimeter long, by what scale factor should you magnify it?

b) If you want to make a photograph of the original paramecium so that its image is 15 centimeters long, by what scale factor should you magnify it?

c) By approximately what scale factor has the paramecium been enlarged to make the image shown?

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Geometry Section 7.3 Notes: Similar Triangles

Example 1: a) Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. b) Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

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Example 2: Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. a) b) Example 3:

a) If ΔRST and ΔXYZ are two triangles such that 23

RSXY

= , which of the following would be sufficient to prove that the triangles are

similar? a) b) c) ∠R ≅ ∠S d) b) Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar?

a) 43

ACDC

= b) m∠A = 2m∠D c) AC BCDC EC

= d) 54

BCDC

=

Example 4: a) Given / /RS UT , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT. b) Given / /AB DE , AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC.

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Example 5: a) Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet. Then he measured the length of Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower? b) On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?

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Geometry Name: _____________________________________ Section 7.3 Practice Worksheet

For numbers 1 and 2, determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning.

1.

2.

For numbers 3 – 6, identify the similar triangles. Then find each measure.

3. LM, QP 4. NL, ML

5. PS, PR 6. EG, HG

7. INDIRECT MEASUREMENT A lighthouse casts a 128-foot shadow. A nearby lamppost that measures

5 feet 3 inches casts an 8-foot shadow. a) Write a proportion that can be used to determine the height of the lighthouse. b) What is the height of the lighthouse?

8. CHAIRS A local furniture store sells two versions of the same chair: one for adults, and one for children. Find the value of x such

that the chairs are similar.

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9. BOATING The two sailboats shown are participating in a regatta. Find the value of x. 10. GEOMETRY Georgia draws a regular pentagon and starts connecting its vertices to make a 5-pointed

star. After drawing three of the lines in the star, she becomes curious about two triangles that appear in the figure, △ABC and △CEB. They look similar to her. Prove that this is the case.

11. SHADOWS A radio tower casts a shadow 8 feet long at the same time that a vertical yardstick casts a shadow half an inch long.

How tall is the radio tower? 12. MOUNTAIN PEAKS Gavin and Brianna want to know how far a mountain peak is from their houses. They measure the angles

between the line of site to the peak and to each other’s houses and carefully make the drawing shown.

The actual distance between Gavin and Brianna’s houses is 112 miles.

a. What is the actual distance of the mountain peak from Gavin’s house? Round your answer to the nearest tenth of a mile.

b. What is the actual distance of the mountain peak from Brianna’s house? Round your answer to the nearest tenth of a mile.

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Geometry Section 7.4 Notes: Parallel Lines and Proportional Parts When a triangle contains a line that is parallel to one of its sides, the two triangles formed can be proved similar using the Angle-Angle Similarity Postulate. Since the triangles are similar, their sides are proportional.

Example 1: a) In , / /RST RT VU∆ , SV = 3, VR = 8, and UT = 12. Find SU. b) In , / /ABC AC XY∆ , AX = 4, XB = 10.5, and CY = 6. Find BY. Example 2:

a) 1 In , = 18, = 36, and = .2

DEF DH HE DG GF∆ Determine whether / /GH FE . Explain.

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b) In ∆WXZ, XY = 15, YZ = 25, WA = 18, and AZ = 32. Determine whether / /WX AY . A midsegment of a triangle is a segment with endpoints that are the _______________________________ of the two sides of the triangle. Every triangle has __________________ midsegments. The midsegments of are , , and .ABC RP PQ RQ∆ Example 3: a) In the figure, DE and EF are midsegments of ΔABC. Find AB, FE, and m∠AFE. b) In the figure, DE and are midsegments of ΔABC. Find BC, DE, and m∠AFD. Example 4: In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in between city blocks. Find the value of x.

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Example 5: Find the values of x and y.

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Geometry Name: _____________________________________ Section 7.4 Practice Worksheet

1. If AD = 24, DB = 27, and EB = 18, find CE.

2. If QT = x + 6, SR = 12, PS = 27, and TR = x - 4,

find QT and TR.

For numbers 3 and 4, determine whether 𝑱𝑲���� ∥ 𝑵𝑴�����. Justify your answer.

3. JN = 18, JL = 30, KM = 21, and ML = 35

4. KM = 24, KL = 44, and NL = 56 JN

For numbers 6 – 9, 𝑱𝑯 ���� is a midsegment of △KLM. Find the value of x.

6. 7.

8. Find x and y. 9. Find x and y.

10. MAPS On a map, Wilmington Street, Beech Drive, and Ash

Grove Lane appear to all be parallel. The distance from Wilmington to Ash Grove along Kendall is 820 feet and along Magnolia, 660 feet. If the distance between Beech and Ash Grove along Magnolia is 280 feet, what is the distance between the two streets along Kendall?

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11. CARPENTRY Jake is fixing an A-frame. He wants to add a horizontal support beam halfway up and parallel to the ground. How long should this beam be?

12. STREETS In the diagram, Cay Street and Bay Street are parallel. Find x. 13. JUNGLE GYMS Prassad is building a two-story jungle gym according to the plans shown. Find x.

14. FIREMEN A cat is stuck in a tree and firemen try to rescue it. Based on the figure, if a fireman climbs to the top of the ladder, how far away is the cat?

15. EQUAL PARTS Nick has a stick that he would like to divide into 9 equal parts. He places it on a piece of grid paper as shown.

The grid paper is ruled so that vertical and horizontal lines are equally spaced. a) Explain how he can use the grid paper to help him find where he needs to cut the stick. b) Suppose Nick wants to divide his stick into 5 equal parts utilizing the grid paper. What can he do?

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Geometry Section 7.5 Notes: Parts of Similar Triangles

Example 1: a) In the figure, ΔLJK ~ ΔSQR. Find the value of x. b) In the figure, ΔABC ~ ΔFGH. Find the value of x.

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Example 2: a) Sanjay’s arm is about 9 times longer than the distance between his eyes. He sights a statue across the park that is 10 feet wide. If the statue appears to move 4 widths when he switches eyes, estimate the distance from Sanjay’s thumb to the statue. b) Use the information from Example 2A. Suppose Sanjay turns around and sees a sailboat in the lake that is 12 feet wide. If the sailboat appears to move 4 widths when he switches eyes, estimate the distance from Sanjay’s thumb to the sailboat.

Example 3: a) Find the value of x. b) Find the value of x.

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Geometry Name: _____________________________________ Section 7.5 Practice Worksheet For numbers 1 – 4, find the value of x.

1. 2. 3. 4. 5. If △JKL ∼ △NPR, 𝐾𝑀����� is an altitude of △JKL, 𝑃𝑇���� is an altitude of △NPR, KL = 28, KM = 18, and PT = 15.75, find PR. 6. If △STU ∼ △XYZ, 𝑈𝐴���� is an altitude of △STU, 𝑍𝐵���� is an altitude of △XYZ, UT = 8.5, UA = 6, and ZB = 11.4, find ZY. 7. PHOTOGRAPHY Francine has a camera in which the distance from the lens to the film is 24 millimeters. a) If Francine takes a full-length photograph of her friend from a distance of 3 meters and the height of her friend is 140

centimeters, what will be the height of the image on the film? (Hint: Convert to the same unit of measure.) b) Suppose the height of the image on the film of her friend is 15 millimeters. If Francine took a full-length shot, what was the

distance between the camera and her friend? 8. FLAGS An oceanliner is flying two similar triangular flags on a flag pole. The altitude of the larger flag is three times the altitude

of the smaller flag. If the measure of a leg on the larger flag is 45 inches, find the measure of the corresponding leg on the smaller flag.

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9. TENTS Jana went camping and stayed in a tent shaped like a triangle. In a photo of the tent, the base of the tent is 6 inches and the altitude is 5 inches. The actual base was 12 feet long. What was the height of the actual tent?

10. PLAYGROUND The playground at Hank’s school has a large right triangle painted in the ground. Hank starts at the right angle

corner and walks toward the opposite side along an angle bisector and stops when he gets to the hypotenuse.

How much farther from Hank is point B versus point A? 11. FLAG POLES A flag pole attached to the side of a building is supported with a network of strings as shown in the figure.

The rigging is done so that AE = EF, AC = CD, and AB = BC. What is the ratio of CF to BE?

12. COPIES Gordon made a photocopy of a page from his geometry book to enlarge one of the figures. The actual figure that he

copied is shown below.

The photocopy came out poorly. Gordon could not read the numbers on the photocopy, although the triangle itself was clear. Gordon measured the base of the enlarged triangle and found it to be 200 millimeters.

a) What is the length of the drawn altitude of the enlarged triangle? Round your answer to the nearest

millimeter. b) What is the length of the drawn median of the enlarged triangle? Round your answer to the nearest

millimeter.

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Geometry Section 7.7 Notes: Scale Drawing and Models A _____________________________________ or ____________________________________ is an object or drawing with lengths proportional to the object it represents. The scale of a model or drawing is the ratio of a length on the model or drawing to the actual length of the object being modeled or drawn. Example 1: a) The distance between Boston and Chicago on a map is 9 inches. If the scale of the map is 1 inch: 95 miles, what is the actual distance from Boston to Chicago? b) The distance between Cheyenne, WY, and Tulsa, OK, on a map is 8 inches. If the scale of the map is 1 inch : 90 miles, what is the actual distance from Cheyenne to Tulsa? The scale factor of a model or drawing is written as a unitless ratio in simplest form. Scale factors are always written so that the model length in the ratio comes first. Example 2: a) A miniature replica of a fighter jet is 4 inches long. The actual length of the jet is 12.8 yards. What is the scale of the model? How many times as long as the actual is the model jet? b) A miniature replica of a fire engine is 9 inches long. The actual length of the fire engine is 13.5 yards. What is the scale of the replica? How many times as long as the model is the actual fire engine? Example 3: a) Gerrard is making a scale model of his classroom on an 11-by-17 inch sheet of paper. If the classroom is 20 feet by 32 feet, choose an appropriate scale for the drawing and determine the drawing’s dimensions. b) Alaina is an architect making a scale model of a house in a 15-by-26 inch display. If the house is 84 feet by 144 feet, what would be the dimensions of the model using a scale of 1 in. : 6 ft?

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Geometry Name: _____________________________________ Section 7.7 Practice Worksheet For numbers 1 – 3, use the map of Central New Jersey shown and an inch ruler to find the actual distance between each pair of cities. Measure to the nearest sixteenth of an inch. 1. Highland Park and Metuchen 2. New Brunswick and Robinvale 3. Rutgers University Livingston

Campus and Rutgers University Cook–Douglass Campus

4. AIRPLANES William is building a scale model of a Boeing 747-400 aircraft. The wingspan of the model is approximately 8 feet

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inches. If the scale factor of the model is approximately 1:24, what is the actual wingspan of a Boeing 747-400 aircraft? 5. ENGINEERING A civil engineer is making a scale model of a highway on ramp. The length of the model is 4 inches. The actual

length of the on ramp is 500 feet. a) What is the scale of the model? b) How many times as long as the actual on ramp is the model? c) How many times as long as the model is the actual on ramp? 6. MOVIES A movie director is creating a scale model of the Empire State Building to use in a scene. The Empire State Building is

1250 feet tall. a) If the model is 75 inches tall, what is the scale of the model? b) How tall would the model be if the director uses a scale factor of 1:75? 7. MONA LISA A visitor to the Louvre Museum in Paris wants to sketch a drawing of the Mona Lisa, a famous painting. The

original painting is 77 centimeters by 53 centimeters. Choose an appropriate scale for the replica so that it will fit on a 8.5-by-11-inch sheet of paper.

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8. MODELS Luke wants to make a scale model of a Boeing 747 jetliner. He wants every foot of his model to represent 50 feet.

Complete the following table.

Part Actual length (in.)

Model length (in.)

Wing Span 2537

Length 2782

Tail Height 392

(Source: Boeing)

9. PHOTOGRAPHS Tracy is 4 feet tall and her father is 6 feet tall. In a photograph of the two of them standing side by side, Tracy’s

image is 2 inches tall. Although their images are much smaller, the ratio of their heights remains the same. How tall is Tracy’s father’s image in the photo? What is the scale of the photo?

10. TOWERS The Tokyo Tower in Japan is currently the world’s tallest self-supporting steel tower. It is 333 meters tall. a) Heero builds a model of the Tokyo Tower that is 2775 millimeters tall. What is the scale of Heero’s model? b) How many times as tall as the actual tower is the model? 11. PUPPIES Meredith’s new Pomeranian puppy is 7 inches tall and 9 inches long. She wants to make a drawing of her new

Pomeranian to put in her locker. If the sheet of paper she is using is 3 inches by 5 inches, find an appropriate scale factor for Meredith to use in her drawing.

12. MAPS Carlos makes a map of his neighborhood for a presentation. The scale of his map is 1 inch:125 feet. a) How many feet do 4 inches represent on the map? b) Carlos lives 250 feet away from Andrew. How many inches separate Carlos’ home from Andrew’s on the map? c) During a practice run in front of his parents, Carlos realizes that his map is far too small. He decides to make his map 5 times as

large. What would be the scale of the larger map?

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CHAPTER 8 Right Triangles & Trigonometry

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Geomety Section 8.1 Notes: Geometric Mean When the means of a proportion are the same number, that number is called the _____________________________ of the extremes. The geometric mean between two numbers is the positive square root of their product.

Example 1: a) Find the geometric mean between 2 and 50. b) Find the geometric mean between 3 and 12. Altitude of a triangle: The altitude of a triangle is a segment drawn from a vertex to the line containing the opposite side and _________________________ to the line containing that side. Geometric Means in Right Triangles: In a right triangle, the altitude drawn from the __________ of the right angle to the _____________________ forms two additional right triangles. These three right triangles share a special relationship.

Example 2: a) Write a similarity statement identifying the three similar triangles in the figure.

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By definition of similar polygons, you can write proportions comparing the side lengths of these triangles.

Notice that the circled relationships involve geometric means. This leads to the next theorem.

Parachute Pete Story *Pete always parachutes from the RIGHT ANGLE of the LARGE triangle. *The path he travels is the GEOMETRIC MEAN (so it is used TWICE in the proportion). *Then he visits TWO cities: (these are the other two blanks in the proportion). If the path he traveled was the MIDDLE path, then he visits the city on the RIGHT or the city on the LEFT. If the path he traveled was an OUTSIDE path, then he visits the CLOSE city or the FAR city.

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Use the letter P for Pete. Path 1: Path 2: Path 3: Example 3: a) Find c, d, and e. b) Find e to the nearest tenth.

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Example 3: Ms. Alspach is constructing a kite for her son. She has to arrange two support rods so that they are perpendicular. The shorter rod is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod? Example 4: A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, AC , intersects BD at point E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot?

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Geometry Name: _____________________________________ Section 8.1 Practice Worksheet Find the geometric mean between each pair of numbers. 1. 8 and 12 2. 3 and 15 3. 4

5 and 2

For numbers 4 and 5, write a similarity statement identifying the three similar triangles in the figure. 4. 5. For numbers 6 – 9, find x, y, and z. 6. 7. 8. 9. 10. CIVIL An airport, a factory, and a shopping center are at the vertices of a right triangle formed by three highways. The airport and

factory are 6.0 miles apart. Their distances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service road will be constructed from the shopping center to the highway that connects the airport and factory. What is the shortest possible length for the service road? Round to the nearest hundredth.

11. SQUARES Wilma has a rectangle of dimensions ℓ by w. She would like to replace it with a square that has the same area. What is

the side length of the square with the same area as Wilma’s rectangle?

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12. EQUALITY Gretchen computed the geometric mean of two numbers. One of the numbers was 7 and the geometric mean turned out to be 7 as well. What was the other number?

13. VIEWING ANGLE A photographer wants to take a picture of a beach front. His camera has a viewing angle of 90° and he wants

to make sure two palm trees located at points A and B in the figure are just inside the edges of the photograph.

He walks out on a walkway that goes over the ocean to get the shot. If his camera has a viewing angle of 90°, at what distance down the walkway should he stop to take his photograph?

14. EXHIBITIONS A museum has a famous statue on display. The curator places the statue in the corner of a rectangular room and builds a 15-foot-long railing in front of the statue. Use the information below to find how close visitors will be able to get to the statue.

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Geometry Section 8.2 Notes: The Pythagorean Theorem and its Converse The Pythagorean Theorem relates the lengths of the hypotenuse and legs of a right triangle.

Example 1: Find the value of x. a) b) A Pythagorean Triple is a set of nonzero whole numbers a, b, and c that satisfy the Pythagorean Theorem. Some common triples are shown in the table below. The triples below are found by multiplying each number in the triple by the same factor.

Example 2: Use a Pythagorean Triple to find the value of x. a) b)

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Example 3: a) A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder? b) A 10-foot ladder is placed against a building. The base of the ladder is 6 feet from the building. How high does the ladder reach on the building? The Converse of the Pythagorean Theorem also holds. You can use this to determine if a triangle is a right triangle if given the measures of all 3 sides.

You can also use the side lengths to classify a triangle as acute or obtuse.

Example 4: a) Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. b) Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer.

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Geometry Name: _____________________________________ Section 8.2 Practice Worksheet For numbers 1 – 3, find x. 1. 2. 3. For numbers 4 – 7, use a Pythagorean Triple to find x. 4. 5. 6. 7. For numbers 8 – 13, determine whether each set of numbers can be measure of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. Justify your answer. 8. 10, 11, 20 9. 12, 14, 49 10. 5√2, 10, 11 11. 21.5, 24, 55.5 12. 30, 40, 50 13. 65, 72, 97 14. CONSTRUCTION The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long. How

high is the dock?

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15. SIDEWALKS Construction workers are building a marble sidewalk around a park that is shaped like a right triangle. Each marble slab adds 2 feet to the length of the sidewalk. The workers find that exactly 1071 and 1840 slabs are required to make the sidewalks along the short sides of the park. How many slabs are required to make the sidewalk that runs along the long side of the park?

16. RIGHT ANGLES Clyde makes a triangle using three sticks of lengths 20 inches, 21 inches, and 28 inches. Is the triangle a right

triangle? Explain. 17. TETHERS To help support a flag pole, a 50-foot-long tether is tied to the pole at a point 40 feet above the ground. The tether is

pulled taut and tied to an anchor in the ground. How far away from the base of the pole is the anchor? 18. FLIGHT An airplane lands at an airport 60 miles east and 25 miles north of where it took off.

How far apart are the two airports?

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Geometry Section 8.3 Notes: Special Right Triangles The diagonal of a square forms two congruent isosceles right triangles. Since the base angles are congruent, the measure of each acute angle is 90 ÷ 2, or 45°. Such a triangle is also known as a 45° – 45° – 90° triangle.

Example 1: Find the measure of each hypotenuse. a) b) You can also work backwards using theorem 8.8 to find the legs of a 45° – 45° – 90° triangle a) b)

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A 30° – 60° – 90° triangle is another special right triangle. You can use an equilateral triangle to find this relationship. When the altitude is drawn from any vertex of an equilateral triangle, two congruent 30° – 60° – 90° triangles are formed. In the figure shown, ∆ 𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷, so 𝐴𝐷���� ≅ 𝐶𝐷.����� If CD = x, then AC = 2x. This leads to the next theorem. Example 2: Find x and y. a) b) c) Find BC. . Example 3: A quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle? Example 4: Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends.

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Geometry Name: _____________________________________ Section 8.3 Practice Worksheet For numbers 1 – 6, find x. 1. 2. 3. 4. 5. 6. For numbers 7 – 10, find x and y. 7. 8. 9. 10. 11. Determine the length of the leg of 45°–45°–90° triangle with a hypotenuse length of 38. 12. Find the length of the hypotenuse of a 45°–45°–90° triangle with a leg length of 77 centimeters. 13. An equilateral triangle has an altitude length of 33 feet. Determine the length of a side of the triangle. 14. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted

in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from a diagonal pathway through the garden. How long is the pathway?

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15. ORIGAMI A square piece of paper 150 millimeters on a side is folded in half along a diagonal. The result is a 45°-45°-90° triangle. What is the length of the hypotenuse of this triangle?

16. ESCALATORS A 40-foot-long escalator rises from the first floor to the second floor of a shopping mall. The escalator makes a

30° angle with the horizontal. How high above the first floor is the second floor? 17. HEXAGONS A box of chocolates shaped like a regular hexagon is placed snugly inside of a rectangular box as shown in the

figure. If the side length of the hexagon is 3 inches, what are the dimensions of the rectangular box? 18. WINDOWS A large stained glass window is constructed from six 30°-60°-90° triangles as

shown in the figure. What is the height of the window?

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Geometry Section 8.4 Notes: Trigonometry The word trigonometry comes from two Greek terms, trigon, meaning triangle, and metron, meaning measure. A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. By AA Similarity, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure. So trigonometric ratios are constant for a given angle measure.

Example 1 a) Express sin L as a fraction and as a decimal to the nearest hundredth. b) Express cos L as a fraction and as a decimal to the nearest hundredth c) Express tan L as a fraction and as a decimal to the nearest hundredth. d) Express sin N as a fraction and as a decimal to the nearest hundredth. e) Express cos N as a fraction and as a decimal to the nearest hundredth. f) Express tan N as a fraction and as a decimal to the nearest hundredth. Special right triangles can be used to find the sine, cosine, and tangent of 30°, 45° and 60° angles. Example 2: a) Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. b) Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth.

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Example 3: A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Example 4: The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch? If you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle, which is the inverse of the trigonometric ratio.

Example 5: a) Use a calculator to find the measure of ∠P to the nearest tenth. b) Use a calculator to find the measure of ∠D to the nearest tenth. Example 6: Solve each right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree. a) b)

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Geometry Name: _____________________________________ Section 8.4 Practice Worksheet For numbers 1 and 2, find sin L, cos L, tan L, sin M, cos M, and tan M. Express each ratio as a fraction. 1. ℓ = 15, m = 36, n = 39 2. ℓ = 12, m = 12√3, n = 24 For numbers 3 – 5, find x. Round to the nearest hundredth. 3. 4. 5. For numbers 6 – 8, use a calculator to find the measure of ∠B to the nearest tenth. 6. 7. 8. 9. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a

vertical rock formation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36°. What is the height of the formation to the nearest meter?

10. RADIO TOWERS Kay is standing near a 200-foot-high radio tower. Use the information in the figure to determine how far Kay

is from the top of the tower. Express your answer as a trigonometric function.

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11. RAMPS A 60-foot ramp rises from the first floor to the second floor of a parking garage. The ramp makes a 15° angle with the ground. How high above the first floor is the second floor? Express your answer as a trigonometric function.

12. TRIGONOMETRY Melinda and Walter were both solving the same trigonometry problem. However, after they finished their

computations, Melinda said the answer was 52 sin 27° and Walter said the answer was 52 cos 63°. Could they both be correct? Explain.

13. LINES Jasmine draws line m on a coordinate plane. What angle does m make with the x-axis? Round your answer to the nearest degree.

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Geometry Section 8.5 Notes: Angles of Elevation and Depression An angle of elevation is the angle formed by a _________________ line and observer’s to an object _________ the horizontal line. An angle of depression is the angle formed by a horizontal line and an observer’s to an object ____________ the horizontal line.

Since horizontal lines are parallel, the angle of elevation and the angle of depression are ___________ by the alternate interior angles theorem. Example 1: a) At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°?

b) At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree? Example 2: Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot?

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Example 3: a) Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon’s position is 154 meters above sea level, and the angles of depression to the two dolphins are 35° and 36°. Find the distance between the two dolphins to the nearest meter.

b) Madison looks out her second-floor window, which is 15 feet above the ground. She observes two parked cars. One car is parked along the curb directly in front of her window and the other car is parked directly across the street from the first car. The angles of depression of Madison’s line of sight to the cars are 17° and 31°. Find the distance between the two cars to the nearest foot.

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Geometry Name: _____________________________________ Section 8.5 Practice Worksheet For numbers 1 and 2, name the angle of depression or angle of elevation in each figure 1. 2. 3. WATER TOWERS A student can see a water tower from the closest point of the soccer field at San Lobos High School. The edge

of the soccer field is about 110 feet from the water tower and the water tower stands at a height of 32.5 feet. What is the angle of elevation if the eye level of the student viewing the tower from the edge of the soccer field is 6 feet above the ground? Round to the nearest tenth.

4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladder reaches a 30-foot roof that needs repair. If

the angle of elevation from the bottom of the ladder to the roof is 55°, how far is the ladder from the base of the wall? Round your answer to the nearest foot.

5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet

on any property. Lindsay wants to determine whether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from the flagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth?

6. GEOGRAPHY Stephan is standing on the ground by a mesa in the Painted

Desert. Stephan is 1.8 meters tall and sights the top of the mesa at 29°. Stephan steps back 100 meters and sights the top at 25°. How tall is the mesa?

7. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean

bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34° and 48°, how far apart are the dogs to the nearest foot?

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8. LIGHTHOUSES Sailors on a ship at sea spot the light from a lighthouse. The angle of elevation to the light is 25°. The light of the lighthouse is 30 meters above sea level. How far from the shore is the ship? Round your answer to the nearest meter.

9. RESCUE A hiker dropped his backpack over one side of a canyon onto a ledge below. Because of the shape of the cliff, he could

not see exactly where it landed. From the other side, the park ranger reports that the angle of depression to the backpack is 32°. If the width of the canyon is 115 feet, how far down did the backpack fall? Round your answer to the nearest foot.

10. AIRPLANES The angle of elevation to an airplane viewed from the control tower at an airport is 7°. The tower is 200 feet high

and the pilot reports that the altitude is 5200 feet. How far away from the control tower is the airplane? Round your answer to the nearest foot.

11. PEAK TRAM The Peak Tram in Hong Kong connects two terminals, one at the base of a mountain, and the other at the summit.

The angle of elevation of the upper terminal from the lower terminal is about 15.5°. The distance between the two terminals is about 1365 meters. About how much higher above sea level is the upper terminal compared to the lower terminal? Round your answer to the nearest meter.

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CHAPTER 9 Transformations & Symmetry

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Geometry Section 9.1 Notes: Reflections In Lesson 4.7, you learned that a reflection or flip is a transformation in a line called the line of reflection. Each point of the preimage and its corresponding point on the image are the same distance from this line. A’, A’’, A’’’ and so on, name corresponding points for one or more transformation. Example 1: Draw the reflected image of quadrilateral WXYZ in line p. Example 2: Suppose that you must bounce the cue ball off side A before it rolls into the pocket at B. Locate the point C along side A that the ball must hit to ensure that it will roll directly toward the pocket. Example 3: Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L(2, –1), and M(0, 1). a) Graph JKLM and its image over x = 1. a) b) b) Graph JKLM and its image over y = –2 .

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When the line of reflection is the x – axis or y – axis, you can use the following rule: Example 4: Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). a) Graph the image reflected in the x – axis. b) Graph the image reflected in the y – axis. \ Example 5: Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph the image under reflection of the line y = x.

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Geometry Name _____________________________________________ Section 9.1 Practice Worksheet For numbers 1 and 2, use the figure and given line of reflection. Then draw the reflected image in this line using a ruler.

1. 2.

For numbers 3 – 6, graph each figure and its image under the given reflection.

3. quadrilateral ABCD with vertices 4. △FGH with vertices F(–3, –1), G(0, 4) A(–3, 3), B(1, 4), C(4, 0), and and H(3, –1) in the line y = x D(–3, –3) in the line y = x

5. rectangle QRST with vertices Q(–3, 2), 6. trapezoid HIJK with vertices H(–2, 5), R(–1, 4), S(2, 1), and T(0, –1) I(2, 5), J(–4, –1), and K(–4, 3) in the x-axis in the y-axis

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7. Vincent is making a star. Complete the star by drawing the reflected image of the figure in line m. 8. Maria placed a paper cutout on a table and then left to get some glue. Her friend James flipped over the cutout without telling Maria. Still, when Maria came back it was impossible for her to be able to see that anything had been changed. Draw the line through the figure that represents the line over which James must have flipped the figure. 9. Tylia drew a shape on a piece of paper and set up a mirror up next to it on her desk. The shape and its image in the mirror are shown below. Draw a line to show where Tylia’s mirror must have been set up. 10. Wilfred hired an interior designer to layout the furniture in his bedroom. The designer produced the plan shown in the figure. Unfortunately, Wilfred’s window is located on the opposite wall from the plan. Wilfred decided to just reflect the plan over the vertical line through the center of the room. Draw the reflected plan. 11. Casey drew this triangle on the coordinate plane: a) What are the vertices of the image of this triangle if it is reflected over the y – axis? b) What are the vertices of the image of this triangle if it is reflected over the line y = x?

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Geometry Section 9.2 Notes: Translations In Lesson 4.7, you learned that a translation or slide is a transformation that moves all points of a figure the same distance in the same direction. Since vectors can be used to describe both distance and direction, vectors can be used to define translations. Example 1: Draw the translation of the figure along the translation vector. Recall that a vector in the coordinate plane can be written as ,a b , where a represents the horizontal change and b is the vertical change from the vector’s tip to its tail. CD is represented by the ordered pair 2, 4− . Written in this form, called the component form, a vector can be used to translate a figure in the coordinate plane.

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Example 2: a) Graph ∆TUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector 3, 2− . b) Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) along the vector 5, 1− − . Example 3: The graph shows repeated translations that result in the animation of the raindrop. a) Describe the translation of the raindrop from position 2 to position 3 in function notation and in words. b) Describe the translation of the raindrop from position 3 to position 4 using a translation vector.

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Geometry Name _____________________________________ Section 9.2 Practice Worksheet For numbers 1 and 2, use the figure and the given translation vector to draw the translation of the figure along the given translation vector. 1. 2. For numbers 3 and 4, graph each figure and its image along the given vector. 3. quadrilateral TUWX with vertices 4. pentagon DEFGH with vertices D(–1, –2), T(–1, 1), U(4, 2), W(1, 5), and X(–1, 3); E(2, –1), F(5, –2), G(4, –4), and H(1, –4); ⟨–2, –4⟩ ⟨–1, 5⟩

For n umbers 5 – 7, find the translation that moves the figure on the coordinate plane. 5. figure 1 figure 2 6. figure 2 figure 3 7. figure 3 figure 4 8. Wynette wants to see the translation of the five-sided figure shown below. Draw the translation so that the indicated vertex is translated to the location of the dot.

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9. A wallpaper design consists of repeated translations of a single isosceles triangle. The pattern is shown overlaid on a coordinate plane. The space above the triangle around the coordinate (5, 1) should be filled with a missing triangle. What are the coordinates of the vertices of the triangle that fill this space consistently with the rest of the pattern? 10. Gus reflects an object twice. The first step is to reflect it over the line y = –1 . Then Gus completes the composite reflection by reflecting it over the line y = 1. The net effect is a translation of the object. Describe this translation. 11. Lacy performs the translation (x, y) (x + 5, y + 3) to an object in the coordinate plane. Kyle performs the translation (x, y) (x – 4, y + 2) to the same object after lacy. What single translation could have been done to achieve the same effect as Lacy and Kyle’s combined translations? Would the result have been different if Kyle did his translation first? 12. Use the coordinate plane to the right to answer parts a and b. a) The image of square S under a translation is square S’. Describe the translation. b) Draw the image of the circle C under the same translation that you described in part a.

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Geometry Section 9.3 Notes: Rotations In Lesson 4.7, you learned that a rotation or turn moves every point of a preimage through a specified angle and direction about a fixed point. The direction of rotation can be either clockwise or counterclockwise. Assume that all rotations are counterclockwise unless stated otherwise. Example 1: Rotate quadrilateral RSTV 45° counterclockwise about point A. When a point is rotated 90°, 180°, or 270° counterclockwise about the origin, you can use the following rules:

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Example 2: Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Graph ΔDEF and its image after a rotation of 115° clockwise about the point G(–4, –2). Example 3: Hexagon DGJTSR is shown below. What is the image of point T after a 90° counterclockwise rotation about the origin? Multiple Choice: a) (5, –3) b) (–5, –3) c) (–3, 5) d) (3, –5) Example 4: Triangle PQR is shown below. What is the image of point Q after a 90° counterclockwise rotation about the origin?

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Geometry Name _____________________________________ Section 9.3 Practice Worksheet For numbers 1 and 2, use the figure and the point of rotation to draw the rotation of the figure. 1. 110° 2. 280° For numbers 3 – 6, graph each figure and its image as a rotation about the origin 3. ∆PQR with P(1, 3), Q(3, –2), and R(4, 2); 90° 4. ∆ABC with A(–4, 4), B(–2, –1), and C(2, –4); 270° 5. quadrilateral WXYZ with W(1, 3), X (3, 1), 6. trapezoid FGHI with F(8, 7), G(5, 8), Y(–5, 6), and Z(–6, 5); 180° H(–3, –7), and I(–7, –2); 90° 7. A damaged compass points northwest. If you travel west by the compass, what is your angle of rotation to true north? 8. Nicki is making a flyer that contains a large capital “M”. She decides that she needs to rotate the “M” clockwise by 60°. Draw the rotated image.

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9. Mariah is looking at her sink. The mark on the faucet for the cold water is shown below. What angle does she need to rotate the faucet so that its mark has the correct orientation? 10. A circular dial with the digits 0 through 9 evenly spaced around its edge can be rotated clockwise 36°. How many times would you have to perform this rotation in order to bring the dial back to its original orientation? 11. A tessellation is when a single shape is repeated to tile a plane with no gaps or overlaps. Below is an example of ∆RST rotated around the point R. This forms a tessellation. Determine if the following shapes can form a tessellation by rotation around the given point. a) b) c)

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Geometry Section 9.4 Notes: Compositions of Transformations When a transformation is applied to a figure and then another transformation is applied to its image, the result is called a composition of transformations. A glide reflection is one type of composition of transformations. Example 1: Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along ⟨5, 0⟩ and a reflection in the x-axis. In Example 1, quadrilateral BGTS ≅ quadrilateral B’G’T’S’ and quadrilateral B’G’T’S’ ≅ quadrilateral B’’G’’T’’S’’. By the Transtiive Property of Congruence, quadrilateral BGTS ≅ quadrilateral B’’G’’T’’S’’. This suggests the following theorem: Example 2: ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along ⟨–1 , 5⟩ and a rotation 180° about the origin.

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The composition of two reflections in parallel lines is the same as a translation. The composition of two reflections in intersecting lines is the same as a rotation. Example 3: Reflect figure EFGH in line p and then line q. Then describe a single transformation that maps EFGH onto E''F''G''H''.

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Example 4: Describe the transformations that are combined to create the brick pattern shown. a) b)

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Geometry Name _____________________________________ Section 9.4 Practice Worksheet For numbers 1 – 4, ∆ABC has vertices A(1, 3), B(–2, –1), and C(3, –2). Graph ∆ABC and its image after the indicated glide reflection. 1. Translation: along ⟨2, 0⟩ 2. Translation: along ⟨–1, 1⟩ Reflection: in y – axis Reflection: in y = x 3. Translation: along ⟨–1, 2⟩ 4. Translation: along ⟨0, 2⟩ Reflection: in x = y Reflection: in y – axis For numbers 5 – 8, reflect figure F in line x and then line y. Describe a single transformation that maps F into F’’. 5. 6. 7. 8.

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9. To make a bridge structurally sound, its base must be a reflection along the center line of the river. Given the base on one side of the river, draw the other base. 10. Describe the transformations that are combined to create the carpet pattern. 11. Surish says a reflection in the x-axis and then a reflection in the y-axis is the same as rotating by 180° about the origin. Thomas

says a reflection in the x-axis and then a reflection in the y-axis is the same as reflecting in the line y = x. Is either of them correct? Explain your reasoning.

12. When a rotation and a reflection are performed as a composition of transformations on an image, does the order of the

transformations affect the location of the final image sometimes, always or never? Explain. 13. Describe the transformations that are combined to create each border. a) b) c)

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Geometry Section 9.5 Notes: Symmetry A figure has symmetry if there exists a rigid motion – reflection, translation, rotation, or glide reflection – that maps the figure onto itself. One type of symmetry is line symmetry. Example 1: State whether the object appears to have line symmetry. Write yes or no. If so, draw all lines of symmetry, and state their number. a) b) c) Another type of symmetry is rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The magnitude of symmetry (or angle of rotation) is the smallest angle through which a figure can be rotated so that it maps onto itself. The order and magnitude of a rotation are related by the following equation:

Magnitude = 360 ÷ order

The figure above has rotational symmetry of order 4 and magnitude 90°.

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Example 2: State whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry, and state the order and magnitude of symmetry. a) b) c) Three-dimensional figures can also have symmetry. Example 3: State whether the figure has plane symmetry, axis symmetry, both, or neither. a) b)

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Geometry Name _____________________________________ Section 9.5 Practice Worksheet For numbers 1 – 3, state whether the figure has line symmetry. Write yes or no. If so, draw all lines of symmetry and state their number. 1. 2. 3. For numbers 4 – 6, state whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry and state the order and magnitude of symmetry.

4. 5. 6. For numbers 7 and 8, state whether the figure has plane symmetry, axis symmetry, both, or neither.

7. 8. 9. A paddle wheel on a steamboat is driven by a steam engine that rotates the paddles attached to the wheel to propel the boat through the water. If a paddle wheel consists of 18 evenly spaced paddles, identify the order and magnitude of its rotational symmetry. 10. Examine each capital letter in the alphabet. Determine which letters have 180° rotational symmetry about a point in the center of the letter. 11. A regular polygon has rotational symmetry with an order of 5 and a magnitude of 72°. What is the figure?

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12. Steve found the hubcaps shown below at his local junkyard. Does each hubcap appear to have line symmetry? 13. Martha made the figure shown. How many lines of symmetry does the figure have? 14. Kelly is designing how she wants to put the place settings for her party. She wants the tables to be set up symmetrically with the

design that is on the table. First she places plates shown. a) What is the order and magnitude of the rotational symmetry of the figure? b) Make the least possible number of the additions of plates to the figure so that the table has rotational symmetry of order 4 around its

center. c) Is it possible to rearrange the locations of the 4 plates in the original figure so that (1) their distance from the center of the table does

not change,(2) their centers remain somewhere on the rectangle design, and (3) the resulting figure has a rotational symmetry of order 4? If so, draw the figure. If not, explain why.

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Geometry Section 9.6 Notes: Dilations A dilation or scaling is a similarity transformation that enlarges or reduces a figure proportionally with respect to a center point and a scale factor. Example 1: Consider trapezoid PQRS and point C. Use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. In Lesson 7.6, you learned that if k > 1, then the dilation is an enlargement. If 0 < k < 1, then the dilation is a reduction. Since 3 is greater than 1, the dilation in Example 1 is an enlargement. A dilation with a scale factor of 1 is called isometry dilation. It produces an image that coincides with the preimage. The two figures are congruent. Example 2: To create the illusion of a “life-sized” image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and the puppet is 24 inches (LP). To what distance PP' should you place the puppet from the screen to create a 49.5-inch tall shadow (I'M') from a 9-inch puppet?

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Use the following rules to find the image of a figure after a dilation centered at the origin: Example 3: Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Graph the image of EFGH after a dilation

centered at the origin with a scale factor of 14

.

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Geometry Name _____________________________________ Section 9.6 Practice Worksheet For numbers 1 and 2, use a ruler to draw the image of the figure after a dilation with center C and scale factor r as indicated.

1. r = 2 2. r = 14

For numbers 3 and 4, determine whether the dilation of K to K’ is an enlargement or reduction. Then find the scale factor of the dilation and solve for x. 3. 4. For numbers 5 and 6, find the image of each polygon with the given vertices after a dilation centered at the origin with the given scale factor. 5. J(2, 4), K(4, 4), P(3, 2); r = 2 6. D(–2, 0), G(0, 2), F(2, –2); r = 1.5 7. Margo superimposed the image of the dilation of a figure on its original figure as shown. Identify the center of this dilation. Explain how you found it.

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8. Tyrone drew a shape together with one of its dilations on the same coordinate plane as shown. What is the scale factor of the dilation? 9. Cara is making images for a poster. She wants to thicken the five pointed star shown by dilating it, and then filling in the space between the original and its image. Sketch the dilated image with the indicated center and a scale factor of 1.5. 10. Leila drew a polygon with coordinates (–1, 2), (1, 2), (1, –2), and (–1, –2). She then dilated the image and obtained another polygon with coordinates (–6, 12), (6, 12), (6, –12), and (–6, –12). What was the scale factor and center of this dilation? 11. Fred drew the footprint of a stage he was planning to build for his band on a coordinate plane. He decided he wanted to make it smaller because he wanted to make sure it fit at every venue. a) Graph the image of Fred’s state after a dilation centered at (0, 0), with scale factor 0.5. b) The perimeter of the image is 26 unites. What is the perimeter of the original figure?