Name Class 9-2
Date Applying Special Right Triangles
Going Deeper
Essential question: What can you say about the side lengths associated with special
right triangles?
Video Tutor
There are two special right triangles that arise frequently in problem-solving
situations. It is useful to know the relationships among the side lengths of
these triangles.
MCC9–12.G.SRT.6
1
EXPLORE
A
Investigating an Isosceles Right Triangle
The figure shows an isosceles right triangle. What is the measure of each base
B
angle of the triangle? Why?
x
B
A
Let the legs of the right triangle have length x. You can use the Pythagorean
C
x
Theorem to find the length of the hypotenuse in terms of x.
​AB​2​= ​x2​ ​+ ​x2​ ​
Pythagorean Theorem
​AB​2​=
© Houghton Mifflin Harcourt Publishing Company
AB =
Combine like terms.
Find the square root of both sides and simplify.
REFLECT
1a. A student claims that if you know one side length of an isosceles right triangle, then
you know all the side lengths. Do you agree or disagree? Explain.
1b. Explain how to find y in the right triangle at right.
B
D
A
Module 9
197
1
y
1
C
Lesson 2
2
MCC9–12.G.SRT.6
EXPLORE
Investigating Another Special Right Triangle
___
A In the figure,
△ABD is an equilateral triangle and ​BC​ is a perpendicular
___
from B to AD​
​  . Explain how to find the angle measures in △ABC.
B Explain why △ACB ≅ △DCB.
B
A
D
C
B
___
___
C Let the length of ​AC​ be x. What is the length of ​AB​ ? Why?
D In the space
below, show how to use the Pythagorean Theorem to find the
___
A
C
x
length of ​BC​.
2a. What is the ratio of the side lengths in a right triangle with acute angles that
measure 30° and 60°?
L
2b. Error Analysis A student drew a right triangle with a 60° angle and a
hypotenuse of length 10. Then he labeled the other side lengths as shown.
Explain how you can tell just by glancing at the side lengths that the student
made an error. Then explain the error.
J
Module 9
198
10 √3
10
60˚
5
K
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
REFLECT
The right triangles you investigated are sometimes called 45°-45°-90° and 30°-60°-90° right
triangles. The side-length relationships that you discovered can be used to find lengths in
any such triangles.
MCC9–12.G.SRT.8
3
E X a m ple
Solving Special Right Triangles
A Refer to the diagram of the 45°-45°-90° triangle. Fill
B
in the calculations that help you find the missing
side lengths. Give answers in simplest radical form.
AC =
BC = AC ·
=
AB = AC ·
=
45°
45°
A
B Refer to the diagram of the 30°-60°-90° triangle. Fill
E
in the calculations that help you find the missing side
lengths. Give answers in simplest radical form.
DE =
DF = DE ÷
EF = DF ·
30°
50
=
=
60°
D
C
C
17
F
Add the side lengths you calculated to the diagrams.
© Houghton Mifflin Harcourt Publishing Company
REFLECT
3a. Suppose you are given the length of the hypotenuse of a 45°-45°-90° triangle. How
can you calculate the length of a leg?
3b. Suppose you are given the length of the longer leg of a 30°-60°-90° triangle. How
can you calculate the length of the shorter leg?
3c. When finding a leg length in a 45°-45°-90° triangle, one student gave the answer ___
​ 30  ​
​√
2 ​
and another gave the answer 15 ​√
2 ​.  Show the answers are equivalent.
Module 9
199
Lesson 2
pr a c t i c e
Find the value of x. Give your answer in simplest radical form.
1.
2.
S
3. A
J
B
45˚
x
x
K
45˚
R
L
14
C
T
3
4.
√2
x
5.M
E
x
N
30˚
6. W
12
5
1
L
30°
x
U
60˚
D
x
F
8. E
7. S
60˚
7
x
F
15
x
6 √3
x
K
30˚
L
x
T
10.Error Analysis Two students were asked to find the value of x in the figure at
right. Which student’s work is correct? Explain the other student’s error.
Roberto’s Work
In a 30°-60°-90° triangle, the
hypotenuse is twice as long as
the shorter leg, so BC = 4. The
ratio of the lengths of the legs
is 1: √
​ 
3 ​,  so x = 4​√
3 ​.
A
Aaron’s Work
In a 30°-60°-90° triangle, the
side lengths are in a ratio of
1: √
​ 
3 ​: 2, so x must
be √
​ 
3 ​ times
___
the length of ​AB​ . Therefore,
x = 8​√
3 ​.
8
60°
B
Module 9
200
x
C
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
M
9.
G
R
V
9-2
Name Class Date __________________
Date Class__________________
Name ________________________________________
Practice
Additional
Practice
5-8
Applying Special Right Triangles
LESSON
Find the value of x in each figure. Give your answer in simplest
radical form.
1.
2.
________________________
3.
_________________________
________________________
Find the values of x and y. Give your answers in simplest radical form.
© Houghton Mifflin Harcourt Publishing Company
4. x = _______ y = _______
5. x = _______ y = _______
6. x = _______ y = _______
Lucia is an archaeologist trekking through the jungle of the Yucatan
Peninsula. She stumbles upon a stone structure covered with creeper
vines and ferns. She immediately begins taking measurements of her
discovery. (Hint: Drawing some figures may help.)
7. Around the perimeter of the building, Lucia finds small alcoves at regular intervals carved
into the stone. The alcoves are triangular in shape with a horizontal base and two sloped
1
equal-length sides that meet at a right angle. Each of the sloped sides measures 14
4
inches. Lucia has also found several stone tablets inscribed with characters. The stone
1
tablets measure 22 inches long. Lucia hypothesizes that the alcoves once held the stone
8
tablets. Tell whether Lucia’s hypothesis may be correct. Explain your answer.
_________________________________________________________________________________________
_________________________________________________________________________________________
7
inches
16
tall. She wonders whether the statues might have been placed in the alcoves. Tell
whether this is possible. Explain your answer.
8. Lucia also finds several statues around the building. The statues measure 9
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
Module
9 Copyright © by Holt McDougal. Additions and changes201
Lesson 2
Original content
to the original content are the responsibility of the instructor.
37
Holt McDougal Geometry
Name ________________________________________ Date __________________ Class__________________
Problem
Solving
Problem Solving
LESSON
5-8
Applying Special Right Triangles
For Exercises 1–6, give your answers in simplest radical form.
1. In bowling, the pins are arranged in a pattern
based on equilateral triangles. What is the
distance between pins 1 and 5?
_________________________________________
2. To secure an outdoor canopy, a 64-inch cord is extended
from the top of a vertical pole to the ground. If the cord
makes a 60° angle with the ground, how tall is the pole?
_________________________________________
Find the length of AB in each quilt pattern.
3.
4.
_________________________________________
________________________________________
Choose the best answer.
6. A shelf is an isosceles right triangle, and
the longest side is 38 centimeters. What
is the length of each of the other two sides?
_________________________________________
________________________________________
Use the figure for Exercises 7 and 8.
Assume UJKL is in the first quadrant, with m∠K = 90°.
7. Suppose that JK is a leg of UJKL, a 45°-45°-90°
triangle. What are possible coordinates of point L?
A (6, 4.5)
C (6, 2)
B (7, 2)
D (8, 7)
8. Suppose UJKL is a 30°-60°-90° triangle and JK
is the side opposite the 60° angle. What are the
approximate coordinates of point L?
F (4.9, 2)
H (8.7, 2)
G (4.5, 2)
J (7.1, 2)
Module
9 Copyright © by Holt McDougal. Additions and changes202
Lesson 2
Original content
to the original content are the responsibility of the instructor.
121
Holt McDougal Geometry
© Houghton Mifflin Harcourt Publishing Company
5. An equilateral triangle has an altitude of
21 inches. What is the side length of
the triangle?