Special Right Triangles ACTIVITY The Community...

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Unit 3 • Similarity, Right Triangles, and Trigonometry 245

ACTIVITYACTIVITY

My Notes

Special Right TrianglesThe Community Quilting ProjectSUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Summarize/Paraphrase/Retell, Activating Prior Knowledge

! e Community Hospital wants to make its rooms more cheerful. ! e hospital asked volunteers to sew quilts for patient rooms and to decorate the common areas. ! e Hoover High Student Council wants to participate in the project. Ms. Jones, a geometry teacher, decides to have her classes investigate the mathematical patterns found in quilts.

Quilts are formed by sewing together three layers of fabric. ! e quilt top is the decorative top layer comprised of quilt blocks. Quilt blocks are o" en squares made up of smaller fabric pieces sewn together to create a pattern. ! e batting is a middle layer of padding. ! e bottom layer is typically a single piece of fabric called the backing. To # nish a quilt, a quilter sews all three layers together using decorative stitches over and through the entire quilt area.

! ere are many di$ erent quilt block designs. O" en these designs are named. ! e quilt block design made up of nine small squares, called the “Friendship Star” is shown.

1. ! e Friendship Star quilt block contains # ve small squares and eight triangles.

a. Identify congruent # gures in the quilt block and explain why they are congruent.

b. Classify the triangles in the quilt block by their angle measures.

c. Classify the triangles in the quilt block by their side lengths.

Some sample quilt block designs are shown below.

3.7

CONNECT TO AP

The special right triangle relationships are used to solve problems in trigonometry and calculus.

245-254_SB_Geom_3-7_SE.indd 245245-254_SB_Geom_3-7_SE.indd 245 2/2/10 11:35:16 AM2/2/10 11:35:16 AM

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246 SpringBoard® Mathematics with Meaning™ Geometry

My NotesMy Notes

The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Marking the Text, Summarize/Paraphrase/Retell, Create Representations, Use Manipulatives

d. What are the measures of the acute angles in each of the triangles? Explain your reasoning.

Part I! e Hoover High Student Council decided to make Friendship Star quilts of various sizes with di" erent size quilt blocks. Small quilts will be made from 3-in. × 3-in. blocks, medium quilts from 4.5-in. × 4.5-in. blocks, and large quilts from 6-in. × 6-in. blocks. Ms. Jones’ classes will explore the dimensions of the triangles in each of these blocks.

2. Reproduce the Friendship Star quilt block design on the grid paper provided by your teacher.

a. Complete the appropriate row in the table below by measuring the dimensions of your entire quilt block and the lengths of the leg and hypotenuse of one triangle in the design you made.

b. Calculate the ratio of the hypotenuse to the leg of the triangle.Write the ratio as a decimal in the appropriate row in the table below.

Dimensions of Quilt Block

Length of Triangle Leg(in inches)

Length of Hypotenuse(in inches)

Ratio of Hypotenuse

to Leg

3 in. × 3 in.

4.5 in. × 4.5 in.

6 in. × 6 in.

c. Gather data from other students to complete the table.

ACTIVITY 3.7continued

Special Right Triangles

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My Notes


Special Right Triangles The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Look for a Pattern, Identify a Subtask, Quickwrite

3. What patterns do you notice in the table in Item 2?

4. Another Friendship Star quilt block is going to have 15-inch sides.

a. How can you determine the length of the legs of the triangleswithout measuring them?

b Using the patterns you observed in the table in Item 3, predict the length of the hypotenuse without measuring it.

c. What is the length of the hypotenuse of the right triangles in this size quilt block?

d. Is your answer to Part(c) the exact length of the hypotenuse?Why or why not?


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My Notes


SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern, Quickwrite

5. Use the Pythagorean ! eorem to " nd the exact values of the missing dimensions and ratios in the table below.

Dimensions of Quilt Block

Length of Triangle Leg(in inches)


Ratio of Hypotenuse

to Leg9 in. × 9 in.

15 in. × 15 in.1

1.52

6 √__

2 4


7. How close are the measured results in the table in Item 2 to the corresponding exact values in the table in Item 5?



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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Quickwrite

8. Suppose that you are only given the length of one leg of an isosceles right triangle (45°– 45°– 90°).

a. Write a verbal rule for ! nding the lengths of the other two sides.

b. Let l be the length of the leg of any isosceles right triangle (45°– 45°– 90°). Use the Pythagorean " eorem to derive an algebraic rule for ! nding the length of the hypotenuse, h, in terms of l.

9. Ms. Jones designed a variation on the Friendship Star quilt block, called the “Twisted Star.” Using the rules you derived in Item 8, what are the dimensions of the smallest triangle on the 12-in. × 12-in. quilt block shown below? Explain how you found your answer.

Special Right TrianglesThe Community Quilting ProjectThe Community Quilting Project

CONNECT TO AP

The special right triangle relationships are used to solve problems in trigonometry and calculus.


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My NotesMy Notes

Special Right Triangles ACTIVITY 3.7continued The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Create Representations, Quickwrite

Ms. Jones introduces her class to a quilt block in the shape of a hexagon. ! is block is formed from six equilateral triangles, divided in half along their altitudes. Ms. Jones’ students know that if an altitude is drawn in an equilateral triangle, two congruent triangles are formed. ! e resulting hexagonal quilt block is shown below.

10. What is the measure of each of the angles in any equilateral triangle?

11. What are the measures of each of the angles in the smallest triangles in the hexagonal quilt block shown above? Explain your answer.


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12. ! e smallest triangles are special scalene right triangles. ! ey are o" en called 30°– 60°– 90° right triangles.

a. How can you determine which leg is shorter and which leg is longer using the angles of the triangle?

b. If each of the sides of the hexagonal quilt block is 4 inches, how long is the shorter leg in the 30°– 60°– 90° right triangle? Explain your answer.

c. What is the relationship between the length of the hypotenuse and the length of the shorter leg in a 30°– 60°– 90° triangle? Explain your answer.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share


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My Notes

Special Right Triangles ACTIVITY 3.7continued The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern

13. Look for patterns between the longer leg of the 30°– 60°– 90° right triangle and the other sides by completing the table below using the Pythagorean ! eorem. Write each ratio in simplest form. Leave all answers in radical form.


Length of Shorter Leg(in inches)

Length of Longer Leg(in inches)

Ratio of Length of Longer Leg to Length

of Shorter Leg684

3.51


15. Suppose that you are only given the length of the shorter leg of a30°– 60°– 90° right triangle.

a. Write a verbal rule for " nding the length of the longer leg.

b. Write a verbal rule for " nding the length of the hypotenuse.


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite

c. Let s be the length of the shorter leg of any 30°– 60°– 90° right triangle. Use the Pythagorean ! eorem to derive an algebraic rule for " nding the length of the longer leg, l, in terms of s.

16. Use your work from Items 14 and 15 to determine the height of the hexagon block shown below, whose sides are 4 inches, if the height is measured from the midpoint of one side to the midpoint of the opposite side. Explain below how you found your answer.


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Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. In a 45°– 45°– 90° triangle, if the length of a leg is 6 cm, the length of the hypotenuse is:

a. 12 cm b. 6 √__

3 cm

c. 6 √__

5 cm d. 6 √__

2 cm

2. Find a and b in the diagram below. ! e triangle is not drawn to scale.

45˚

b

a

7√2 cm

3. Square MNOP has a diagonal of 12 inches. Find the length of each side of the square.

4. Find a and b.

30˚

42

a

b

5. Find a and c.

10

a

c60˚

6. Find m.

8

60˚

m

a. 4 b. 4 √__

2

c. 4 √__

3 d. 8 √__

3

7. A ladder leaning against a house makes an angle of 60° with the ground. ! e foot of the ladder is 7 feet from the house. How long is the ladder?

8. MATHEMATICAL R E F L E C T I O N

Brayden’s teacher asked him to draw a 30°–60°–90°

right triangle. He drew the diagram shown. Tell why it is not possible for Brayden’s triangle to exist.

512

13

60˚30˚



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