Special Right Triangles
The Community Quilting Project
SUGGESTED LEARNING STRATEGIES: Close Reading, Marking
the Text, Summarize/Paraphrase/Retell, Activating Prior
Knowledge
© 2010 College Board. All rights reserved.
The Community Hospital wants to make its rooms more cheerful.
The hospital asked volunteers to sew quilts for patient rooms and to
decorate the common areas. The 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. The quilt
top is the decorative top layer comprised of quilt blocks. Quilt blocks are
often squares made up of smaller fabric pieces sewn together to create
a pattern. The batting is a middle layer of padding. The bottom layer is
typically a single piece of fabric called the backing. To finish a quilt, a
quilter sews all three layers together using decorative stitches over and
through the entire quilt area.
There are many different quilt block designs. Often these designs are
named. The quilt block design made up of nine small squares, called the
“Friendship Star” is shown.
ACTIVITY
3.7
My Notes
Some sample quilt block
designs are shown below.
1. The Friendship Star quilt block contains five small squares and
eight triangles.
a. Identify congruent figures in the quilt block and explain why they
are congruent.
b. Classify the triangles in the quilt block by their angle measures.
CONNECT TO AP
c. Classify the triangles in the quilt block by their side lengths.
The special right triangle
relationships are used to solve
problems in trigonometry and
calculus.
Unit 3 • Similarity, Right Triangles, and Trigonometry
245
ACTIVITY 3.7
continued
Special Right Triangles
The Community Quilting Project
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Summarize/Paraphrase/Retell, Create Representations,
Use Manipulatives
My Notes
d. What are the measures of the acute angles in each of the triangles?
Explain your reasoning.
Part I
The Hoover High Student Council decided to make Friendship Star quilts
of various sizes with different 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.
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.
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SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
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.
Special Right Triangles
ACTIVITY 3.7
continued
The Community Quilting Project
SUGGESTED LEARNING STRATEGIES: Predict and Confirm,
Look for a Pattern, Identify a Subtask, Quickwrite
My Notes
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 triangles
without measuring them?
b Using the patterns you observed in the table in Item 3, predict the
length of the hypotenuse without measuring it.
© 2010 College Board. All rights reserved.
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?
Unit 3 • Similarity, Right Triangles, and Trigonometry
247
ACTIVITY 3.7
continued
Special Right Triangles
The Community Quilting Project
My Notes
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Create Representations, Look for a Pattern, Quickwrite
5. Use the Pythagorean Theorem to find the exact values of the missing
dimensions and ratios in the table below.
Dimensions of
Quilt Block
Length of
Triangle Leg
(in inches)
Length of
Hypotenuse
(in inches)
Ratio of
Hypotenuse
to Leg
9 in. × 9 in.
15 in. × 15 in.
1
1.5
2
__
6√ 2
4
6. What patterns do you notice in the table in Item 7?
© 2010 College Board. All rights reserved.
7. How close are the measured results in the table in Item 2 to the
corresponding exact values in the table in Item 5?
248
SpringBoard® Mathematics with Meaning™ Geometry
Special Right Triangles
ACTIVITY 3.7
continued
The Community Quilting Project
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Create Representations, Quickwrite
My Notes
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 finding 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 Theorem to derive an
algebraic rule for finding the length of the hypotenuse, h, in
terms of l.
CONNECT TO AP
The special right triangle
relationships are used to solve
problems in trigonometry and
calculus.
© 2010 College Board. All rights reserved.
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.
Unit 3 • Similarity, Right Triangles, and Trigonometry
249
ACTIVITY 3.7
continued
Special Right Triangles
The Community Quilting Project
My Notes
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.
This 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. The resulting
hexagonal quilt block is shown below.
10. What is the measure of each of the angles in any equilateral triangle?
© 2010 College Board. All rights reserved.
11. What are the measures of each of the angles in the smallest triangles
in the hexagonal quilt block shown above? Explain your answer.
250
SpringBoard® Mathematics with Meaning™ Geometry
Special Right Triangles
ACTIVITY 3.7
continued
The Community Quilting Project
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Think/Pair/Share
My Notes
12. The smallest triangles are special scalene right triangles. They are
often 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.
© 2010 College Board. All rights reserved.
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.
Unit 3 • Similarity, Right Triangles, and Trigonometry
251
ACTIVITY 3.7
continued
Special Right Triangles
The Community Quilting Project
My Notes
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 Theorem. Write each ratio in simplest form. Leave all
answers in radical form.
Length of
Hypotenuse
(in inches)
6
Length of
Shorter Leg
(in inches)
Length of
Longer Leg
(in inches)
Ratio of Length of
Longer Leg to Length
of Shorter Leg
8
4
3.5
1
14. What patterns do you notice in the table in Item 13?
15. Suppose that you are only given the length of the shorter leg of a
30°– 60°– 90° right triangle.
b. Write a verbal rule for finding the length of the hypotenuse.
252
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
a. Write a verbal rule for finding the length of the longer leg.
Special Right Triangles
ACTIVITY 3.7
continued
The Community Quilting Project
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Quickwrite
My Notes
c. Let s be the length of the shorter leg of any 30°– 60°– 90° right
triangle. Use the Pythagorean Theorem to derive an algebraic rule
for finding the length of the longer leg, l, in terms of s.
© 2010 College Board. All rights reserved.
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.
Unit 3 • Similarity, Right Triangles, and Trigonometry
253
Special Right Triangles
ACTIVITY 3.7
continued
The Community Quilting Project
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper. Show your work.
Write
paper.
6. Find m.
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:
__
b. 6 √3 cm
c. 6 √5 cm
d. 6 √2 cm
8
__
__
m
2. Find a and b in the diagram below. The
triangle is not drawn to scale.
a. 4
a
45˚
b
60˚
__
b. 4 √2
__
__
c. 4 √3
d. 8 √3
7. A ladder leaning against a house makes an
angle of 60° with the ground. The foot of the
ladder is 7 feet from the house. How long is
the ladder?
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.
8. MATHEMATICAL Brayden’s teacher asked
R E F L E C T I O N 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.
42
a
5
30˚
12
60˚
b
13
5. Find a and c.
a
c
254
10
60˚
SpringBoard® Mathematics with Meaning™ Geometry
30˚
© 2010 College Board. All rights reserved.
a. 12 cm