math 3 plane geometry part 1 review & 3d · math 3 pg part 1 review & 3d june 5, 2017 area...

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Math 3 PG Part 1 Review & 3D June 5, 2017

Math 3 Plane Geometry Part 1 Review & 3D

MATH 1 REVIEW: PERIMETER AND AREA

1. What is the area of parallelogram EFGH?

2. What is the perimeter of parallelogram EFGH?

3. In the figure to the right, sides a, b, c, d are 8 inches, 12 inches, 10

inches, and 9 inches respectively. The height is 8 inches. What is the

area of the trapezoid?

4. A rectangular rug has an area of 63 square feet, and its width is exactly 2 feet shorter than its length.

What is the length, in feet, of the rug?

5. What is the area in square units of the following figure?

6. In the following figure, AB is perpendicular to BC. The lengths of AB and BC, in inches, are given in terms of x. Which of the following represents the area of triangle ABC, in square inches, for all x > 1? A. 𝑥2 − 1

B. 2𝑥 C. 𝑥2

D. 𝑥2− 1

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E. Cannot be determined from information given

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AREA AND CIRCUMFERENCE OF CIRCLES Area of a circle = 𝝅𝒓𝟐 Circumference of a circle = 𝟐𝝅𝒓

7. The figure below shows 4 congruent circles, each tangent to 2 other circles and to 2 sides of the

square. If the perimeter of the square is 8 cm, then what is the area, in square cm, of 1 circle?

8. In the figure below, the top and bottom of the rectangle are tangent to the circle as shown. The

rectangle has a length of 4𝑥 and width of 2𝑥 . What is the area of the shaded region?

A. 8𝜋2 + 4𝑥 B. 8𝑥2 + 8𝜋 C. 8𝑥 - 16𝜋2

D. 8𝑥2 - 𝜋𝑥 E. 8𝑥2 - 𝜋𝑥2

9. In the figure below, the circumference of circle X is 10 𝜋 and the circumference of circle Y is 6 𝜋. What

is the length of XY?

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PYTHAGOREAN THEOREM Pythagorean theorem 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐

10. What is the length, in inches, of the diagonal of a rectangle whose dimensions are 16 inches by 30

inches?

11. What is the height of the trapezoid at right?

INTERIOR ANGLES OF A TRIANGLE : The three interior angles of any triangle add up to 180°

12. In the figure below, A, D, B, and G are collinear. If angle CAD measures 70°, angle BCD measures 45°,

and angle CBG measures 130°, what is the degree measure of angle ACD?

13. In triangle RST, angle R is a right angle and angle S measures 30°. Find angle T.

14. The isosceles triangle in the quilt square at right has one angle that

measures 40°. What is the measure of each of the other two angles in the

triangle?

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PARALLEL LINES, TRANSVERSALS, AND CORESPONDING ANGLES A line that cuts through two parallel lines is called a transversal. The angles in matching corners are called corresponding angles. In this figure angle a corresponds with angle e. Angle b corresponds with angle f and so on. Corresponding angles are equal (congruent). And since vertical angles are equal, that means that angles a, c, e, and g are all equal. Likewise angles b, d, f, and h are also equal. And since angle a and angle b form a line, the sum of the angles is 180°.

15. In the figure to the right. Lines l and m are parallel. Find the measures

of angles, 1, 2, 3, 4, 5, 6, and 7.

16. In the figure at right, line l is parallel to line m. Transversals

t and u intersect at point A on l and intersect m at points C and B, respectively. Point X is on m, the measure of angle ACX is 130°, and the measure of angle BAC is 80°. How many of the angles formed by rays of l, m, t, and u have measure 50°?

17. In the following figure, line t crosses parallel lines m and n. If 𝑎° = 130 what is 𝑏° + 𝑑°?

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MATH 2 REVIEW: INTERIOR ANGLES OF A POLYGON If n is the number of sides of the polygon, then the sum of the measures of the interior angles is (𝒏 − 𝟐) × 𝟏𝟖𝟎°

18. What is the sum of the interior angles of this irregular octagon?

19. What is the measure of angle x in the following figure?

20. The following figures show regular polygons and the sum of the degrees of the angles in each polygon.

Based on these figures, what is the number of degrees in an n-sided regular polygon?

A. 180(𝑛 − 2)

B. 180𝑛

C. 60𝑛 D. 20𝑛2 E. Cannot be determined from the information given.

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DIAGONALS: A polygon's diagonals are line segments from one corner to another. Diagonals of certain

quadrilateral have special properties.

Quadrilateral Mutually bisecting

diagonals

Perpendicular

diagonals

Perpendicular

bisecting diagonals

Square Yes Yes Yes

Rhombus Yes Yes Yes

Rectangle Yes No No

Parallelogram Yes No No

Kite No Yes No

Trapezoid No No No

21. In parallelogram ABCD at right, BD and AC are

diagonals which intersect at point E. Length AC is 12

cm and length BD 10 cm. What is the length of BE?

22. In the rhombus at right, diagonal AC = 6 and diagonal BD = 8. What

is the length of each of the four sides?

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CIRCLE ARC LENGTH AND SECTOR AREA

"Arc length" is a fraction of

the circle's circumference:

𝒙

𝟑𝟔𝟎 × circumference

"Sector area" is a fraction of

the circle's area

𝒙

𝟑𝟔𝟎 × area

23. The youth center has installed a swimming pool on level ground. The pool is a right circular cylinder

with a diameter of 30 feet and a height of 6 feet. A diagram of the pool and its entry ladder is shown

below and to the left. A plastic cover is made for the pool. The cover will rest on the top of the pool

and will include a wedge-shaped flap that forms a 45° angle at the center of the cover, as shown in the

figure below and to the right. A zipper will go along one side of the wedge-shaped flap and around the

arc. What is the length of the zipper, in feet rounded to the nearest tenth of a foot?

24. Sam ordered a 14" (diameter) pizza. He ate 2 slices leaving an empty space with a 120° central angle.

How much pizza, measured in square inches, did Sam eat? (round to the nearest tenth)

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SIMILAR TRIANGLES Corresponding angles are equal and corresponding sides are proportional. The triangles (at right) are similar because they have the same angles. That

means that the corresponding sides are proportional. To find the missing

value of s in the figure above, set up a proportion. Since side bc of the first

triangle is proportional to side bc of the second triangle and side ab of the

first triangle is proportional to side ab of the second triangle then

3

4 =

6

𝑠 . Solve for s by cross multiplying. 4 x 6 = 3s, therefore s = 8

25. In the figure below, angle ABC is a right angle and DF is parallel to AC. If AB is 12 inches long, BC is 9

inches long. AD is 6 inches long. What is the area, in square inches, of triangle DBF?

26. In the figure below, angle M is congruent to angle K, angle N and angle L are right angles. Solve for u.

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27. In the figure at right, QS is parallel to PT. The length of RS is 2

and the length of ST is 6. The perimeter of PRT is 24 inches.

What is the perimeter of QRS?

28. For the triangles in the figure at right, which of the following ratios of

side lengths is equivalent to the ratio of the perimeter of ∆ABC to the

perimeter of ∆DAB?

A. AB:AD B. AB:BD C. AD:BD D. BC:AD E. BC: BD

SIMILAR POLYGONS Similar polygons are also proportional, so you can set up a ratio to solve for missing information. All squares are similar. All circles are similar.

29. Given parallelogram ABCD below and parallelogram EFGH (not shown) are similar, which of the

following statements must be true about the two shapes?

A. Their areas are equal. B. Their perimeters are equal. C. Side AB is congruent to side EF. D. Diagonal AC is congruent to diagonal EG. E. Their corresponding angles are congruent.

30. Rectangles ABCD and EFGH shown are similar. Using the given information, what is the length of side

FG, to the nearest tenth of an inch?

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VOLUME AND SURFACE AREA Know how to calculate volume and surface area of various shapes when the formula is given.

31. The volume of a cone, which is derived by treating it as a pyramid with infinitely

many lateral faces, is given by the formula V = 𝟏

𝟑𝝅𝒓𝟐𝒉, where r is the radius of the

base, and h is the height. If the radius is 4 and the height is 8 what is the volume of

the cone? Leave answers in terms of 𝜋.

32. If the volume of a sphere is 288𝜋 cubic inches, then what is the surface area,

in square inches of the same sphere? (Note: for a sphere with radius r, the

volume is 𝟒

𝟑𝝅𝒓𝟑 and the surface area is 𝟒𝝅𝒓𝟐)

DIAGONAL OF A CUBE 𝒅 = 𝒔√𝟑 You can find the diagonal of a cube in one of two ways: remembering the formula or working it out by using using the Pythagorean theorem twice. In the diagram below, line b is a diagonal of one of the sides. This diagonal makes the base leg of a new triangle with hypotenuse d, which is the diagonal of the cube. The formula for finding the diagonal of any

rectangular solid is d = √𝒍𝟐 + 𝒃𝟐 + 𝒉𝟐 where 𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑏 =𝑏𝑟𝑒𝑎𝑑𝑡ℎ, 𝑎𝑛𝑑 ℎ = ℎ𝑒𝑖𝑔ℎ𝑡. In this case, since it's a cube and the length,

breadth, and height are all the same 𝑑 = √3𝑠2 or 𝑑2 = 3𝑠2 and if you take

the square root of both sides √𝑑2 = √3𝑠2 so 𝒅 = 𝒔√𝟑

33. A sphere is inscribed in a cube with a diagonal of 3√3 ft. In feet, what is the

diameter of the sphere?

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GEOMETRIC SEQUENCE - means a pattern of multiplication, in which each term is multiplied by the same

amount to determine the next term. In a geometric sequence, each term divided by the prior term yields a

constant, or common, ratio. For example 3, 6, 9, 12, 15... is a geometric sequence since each term is

multiplied by 3 to get the next term. The Fibonacci sequence is an example of a geometric sequence where

each new number is approximately 1.6 times the previous number.

34. The degree measures of the 4 angles of quadrilateral LMNO,

shown at right, form a geometric sequence with a common

ratio of 2. What is the measure of angle N?

Arithmetic sequence

An arithmetic sequence goes from one term to the next by adding the same number each time. For example,

1, 3, 5, 7, 9, 11, ... is an arithmetic sequence since we add 2 each time to get the next term in the sequence.

The angles in a 30°-60°-90° triangle follow a arithmetic sequence. The smallest angle is 30°, then add 30° for

the next angle and add 30° again to get the measure for the last angle.

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MATH 3 RELATIONSHIPS BETWEEN TWO DIMENSIONAL AND THREE DIMENSIONAL OBJECTS Slicing or cutting through a three dimensional figure with a plane can create a two dimensional shape. For instance slicing through a cone can create a triangle, circle, parabola or ellipse.

Slicing a cone through the vertex

creates a triangle (left)

Slicing a cone diagonally, through the

base, creates a parabola (right)

Slicing a cone parallel to the base

creates a circle (left)

Slicing a cone diagonally creates an

ellipse (right)

Whenever a slice is made parallel to the base of the three-dimensional object then the two-dimensional cross-

section created will be similar to the base. Additionally, the maximum number of sides that a two-dimensional

cross-section can have is equal to the number of faces of the three-dimensional figure from which it is sliced.

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For questions 35-40 determine the two-dimensional cross-section that is created from each slice described

35. Horizontal slice through a right cylinder

36. Vertical slice through a right cylinder

37. Diagonal slice (not through a base) through a right cylinder

38. Horizontal slice through a square based pyramid

39. Vertical slice through the vertex opposite the base through a square based pyramid

40. Vertical slice not through the vertex opposite the base of a square

based pyramid

Rotating a two-dimensional figure around an axis creates a three-dimensional figure.

Start with a rectangle that has a side touching both

the x and y axis. (left)

Rotating the rectangle around the x-axis creates a

right circular cylinder with a height x and radius y.

(right)

Rotating the rectangle around the y-axis creates a

right circular cylinder with a height y and radius x.

(left)

Rotating a rectangle that has only one side on an axis

creates a cylinder with a hole in the middle or a

doughnut. (right)

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For questions 41 - 44 Sketch the result of rotation each shape around the given axis.

41.

42.

43.

44.

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Answers

1. 240 𝑢𝑛𝑖𝑡𝑠2 2. 66 units 3. 80 𝑖𝑛𝑐ℎ𝑒𝑠2 4. 9 feet 5. 104 𝑢𝑛𝑖𝑡𝑠2 6. D

7. 1

4𝜋𝑐𝑚2 or 0.785 𝑐𝑚2

8. E 9. 8 10. 34 inches 11. 3 12. 15° 13. 60° 14. 70° 15. 135°, 45°, 135°, 45°, 45°, 45°, 135° 16. 8 17. 100° 18. 1080° 19. 120° 20. A 21. 5 cm 22. 5 23. 26.8 ft 24. 51.3 𝑖𝑛𝑐ℎ𝑒𝑠2 25. 13.5 𝑖𝑛𝑐ℎ𝑒𝑠2 26. 6 27. 6 inches 28. A 29. E

30. 16

3 or 5.333

31. 128

3𝜋

32. 144𝜋

33. 3 feet

34. 96°

35. circle

36. rectangle

37. ellipse

38. square

39. triangle

math 3 plane geometry part 1 review & 3d · math 3 pg part 1 review & 3d june 5, 2017 area...

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