Breakout Session #52D and 3D geometry

Presented byDr. Del FersterA great deal of time today will be spent doing problems. Let’s start with these.

First on the agenda:A Set of problems dealing with 2 dimensional Geometry!

Key Ideas: Circles: Angles and Arcs; Circles: Sectors and Segments; Coordinates on the x-y plane, Distance Formula, Midpoint, Surface Area

CIRCLES

› The vertex of a central angle is at the center of the circle.

› is a CENTRAL ANGLE

› The measure of a central angle IS EQUAL TO the measure of the arc that it intercepts.

A review of central angles

A

D

B

M

V

AMB»AMB mAB

CIRCLES

› The vertex of an inscribed angle is a point on the circle.

› is an INSCRIBEDANGLE

› The measure of an inscribed angle IS EQUAL TO HALF the measure of the arc that it intercepts.

A review of inscribed angles

ABC»1

2ABC mAC

CIRCLESNow, a quick look at

SECTORS

And

SEGMENTS

Area of a SECTOR of a circle

m˚

360πr2Area of a sector =

measure of the central angle or arc

The fraction of the circle

The area of the entire circle

.

› To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

Finding the area of a segment

Now: Some formulas to help us along.

Distance Formula

Midpoint Formula

2 21 2 1 2( ) ( )d x x y y

1 2 1 2midpoint ,2 2

x x y y

More Formulas Shape Picture Formula for Volume

Rectangular Solid

Surface Area(open box--no lid)

V l w h

2 2SA lh wh lw

Problem #1

› For the circle shown to the right, with center A, and , find the measure of

100OBAC BDC

Problem #2

› The drawing to the right shows a circle with radius 8 inches, and a central angle of . Calculate the area of the shaded sector. You may leave your answer in terms of or use 3.14 for or use the key on your calculator.

60o

Problem #3› For the circle shown to

the right, BC is a chord of the circle. The center of the circle is A, and the radius of the circle is 10 inches. . Calculate the shaded area of the segment to the nearest tenth of a square inch. Use 3.14 for or use the key on your calculator.

90OBAC

Problem #4

› You have just plotted the following points:A: (1,-1)B: (4,3)C: (0,3)D: (-3,-1)

› Determine the perimeter of quadrilateral ABCD.

x

y

Problem #5

› The drawing at the right shows the line with equation

› Points A and B are the y-intercept and x-intercept of the line.

› Point M is the midpoint of

› Determine the coordinates of point M.

x

y

5 3 30x y

A

M

B

AB

Problem #6

› A rectangular prism (Think shoe box) has dimensions 4 inches by 5 inches by 8 inches. Find the surface area of the rectangular prism.

Problem #7

› Sam is going to visit his friend Sally. He decides that he’ll stop by Sheetz to pick up a tasty treat, and then walk past 7-11 on the way to Sally’s house. On the way home, he’ll take the shortest path home-the one past the Wawa.

x

y

SAM

SHEETZ

7-11

SALLY

WAWA

Problem #7 (Continued)› A. If every unit on the

grid corresponds to one block, how many blocks will Sam walk on the way

to Sally’s house?

› B. Again, assuming that each unit on the grid corresponds to one

block, how many blocks will he walk on the way home from Sally’s house?

x

y

SAM

SHEETZ

7-11

SALLY

WAWA

Problem #7 (Concluded)

› C. As Sam is walking home, he decides to stop at Wawa (hey, the coffee might still be on sale for $1!). Wawa is exactly half way from Sally’s house to his house. Give the coordinates for Wawa.

x

y

SAM

SHEETZ

7-11

SALLY

WAWA

Problem #8› Stella is making a case to hold her beanie babies (seems

Stella is stuck in the 90s ). She decides that the bottom of the case needs to be made from regular plywood that she can buy from Lowes at a cost of $.10 per square inch. The sides of the case are going to be made from decorative linoleum, that Stella has chosen to reflect her eclectic artistic tastes. She will buy this material from a local builder at a cost of $.20 per square inch. Stella determines that the base of her case must be a square that measures 20 inches per side. She decides that the sides of the case must be 30 inches high. She will NOT put a lid on her case—after all, beanie babies must be fee to enjoy the view of the world!

Problem #8 (Concluded)

› A. Determine the area of the bottom of the case.

› B. Determine the area of each side of her case.

› C. Determine the surface area of her case. Remember, there is no lid.

› D. Determine the volume of her case.

› E. Use the prices in the previous slide to determine the cost of the materials needed to make the case.

Problem #9

› Find the perimeter of triangle ABC shown at the right.

x

y

C

B

A

Problem #10

› Sparky is wrapping 2 lovely gifts for his math teacher in Green Bay Packers wrapping paper. GO PACK GO!! Each gift has the shape of a rectangular prism (Think shoe box). The larger of the 2 boxes has a length, width, and height that is exactly 3 times as large as the corresponding measurements of the smaller box. What is the relationship between the surface area of the 2 gift boxes?

Problem #11› As shown, a goat is

tied to the corner of a 5 meter by 4 meter rectangular shed by an 8 meter piece of rope. Rounded to the nearest square meter, calculate the area upon which the goat is able to graze. Use 3.14 for

Goat is tied here

Rope reaches to here

Rope reaches to here

5 meters

4 meters

Next up:Some problems that involve 3dimensional Geometry!Key Ideas: Volume of cone, volume of sphere, Volume of a cylinder, Volume of a Rectangular solid.

First, though, a picture of Conor!

First: Some formulas to help us along. Shape Picture Formula for Volume

Cone

Sphere

21

3V r h

34

3V r

More Formulas Shape Picture Formula for Volume

Cylinder

Rectangular Solid

2V r h

V l w h

Problem #12

› Find the volume of a cone with radius 6 cm. and height 10 cm. Round your answer to the nearest tenth of a square cm. Use 3.14 for , or use the key on your calculator.

Problem #13› A solid plastic toy is made in

the shape shown to the right (a cylinder which is joined to a hemisphere at both ends). The diameter of the toy at the joints is 5 cm. The length of the cylindrical part of the toy is 10 cm. Calculate the volume of plastic needed to make the toy. Round your answer to the nearest tenth of a cubic centimeter. Use 3.14 for or use the key on your calculator.

Problem #14

› Calculate the volume of the triangular prism shown to the right.

3 cm

4 cm5 cm

7 cm

Problem #15› Del is wrapping gift boxes in beautiful wrapping paper.

Each gift box is a rectangular prism (think shoe box). The larger of the two boxes has a length, width, and height that is three times as large as the corresponding measurements of the smaller box. What is the relationship between the volumes of the 2 gift boxes?

Problem #16

› Ice cream is sold in stores in cylindrical containers, like the one shown to the right. The containers are 20 cm high with a radius of 8 cm.

› A. Determine how much ice

cream is in each container.

(Hint: think Volume)

Problem #16 (Continued)› Ice cream is sold in stores in

cylindrical containers, like the one shown to the right. The containers are 20 cm high with a radius of 8 cm.

› B. Each scoop of ice cream is a sphere of radius 3 cm. Determine how much ice cream is in each scoop.

(Hint: think Volume)

Problem #16 (Continued)› Ice cream is sold in stores

in cylindrical containers, like the one shown to the right. The containers are 20 cm high with a radius of 8 cm.

› C. How many scoops are in each

container?

Problem #16 (Concluded)› Ice cream is sold in stores in

cylindrical containers, like the one shown to the right. The containers are 20 cm high with a radius of 8 cm.

› D. If one scoop sells for $3.50, how much money will the ice cream store make for each full container of ice cream that it sells in cones?

Solutions› I’ve generated a solution key for these problems that I’ll post on the WIKI WIKI WIKI

› If you want a copy tonight, just let me know.

And now, the teacher stuff!

› I have brought several handouts, that you are welcome to have. Some are practice problem types, dealing with all of the concepts that we’ve considered in this PowerPoint.

› Other handouts are more suitable for use with your students.

› As always, you’re welcome to have copies of anything that you wish.

Wrapping Up

›Thanks for your attention and participation.› In the midst of our seemingly endless amount of snow days, 2 hour delays, and disruptions to schedules, I’m fully aware of just how tough the days are!

›Hang in there. I have the utmost respect for what you do as a professional!

› If I can help in any way, don’t hesitate to shoot me an email or give me a call.