Lesson 18
TAKS Grade 8 Objective 3
(8.7)(C)
The Pythagorean Theorem
The legs of a right triangle are the two shorter sides. The hypotenuse of a
right triangle is the longest side. The hypotenuse is always opposite the right
angle.
a
The Pythagorean Theorem states that, in any right
triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
hy
po
ten
c use
Looking at the diagram: c2 a2 b2
b
New Vocabulary
• legs
• hypotenuse
• Pythagorean
Theorem
Understanding the Pythagorean Theorem
If you make each side of a right triangle into the side of a square, you will
notice that the sum of the areas of the two smaller squares is equal to the
area of the largest square.
A right angle is a 90
angle.
A right triangle is any
triangle with a right
angle.
a. Find the length of the hypotenuse of a right triangle whose legs
are 4 cm and 8 cm.
Step 1 Use the Pythagorean Theorem.
a2 b2 c2
(4 cm)2 (8 cm)2 c2
Step 2 Solve for c.
c 8.9 cm
b. You can also use the Pythagorean Theorem
to find the length of a leg.
(5 cm)2 b2 (10 cm)2
25 cm2 b2 100 cm2
b2 75 cm2
10 cm
b 8.7 cm
8 cm
4 cm
Remember to add a2
and b2 before finding
the square root to
determine c. Although
a2 b2 c2,
a b c.
5 cm
Quick Check 1
1a. Find, to the nearest 0.1 cm, the length of
the hypotenuse of a right triangle whose
legs are 7 cm and 9 cm.
52
LESSON 18
■
The Pythagorean Theorem
1b. A right triangle has legs of the same length
and a hypotenuse 16 cm long. Find the
length of a leg, to the nearest 0.1 cm.
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
EXAMPLE 1
TAKS Objective 3 (8.7)(C)
LESSON 18
Applying the Pythagorean Theorem
The Pythagorean Theorem can be applied to a wide variety of problems.
It can be used to find lengths that would be difficult to measure directly.
EXAMPLE 2
Suppose you are surveying a particularly marshy piece of land. The mud is so thick you
cannot drive across it. You find that the distance from point A to one end of the marsh is
2.1 km. You return to point A and drive at a right angle 1.7 km to the other end of the
marsh. How long is the marsh?
A
2.1 km
1.7 km
marsh
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
Step 1 Identify the legs and hypotenuse of the right triangle.
The distances you measured were at right angles to each
other. Since the hypotenuse is always opposite the right
angle, the distance across the marsh is the hypotenuse.
The legs are the distance you measured.
Step 2 Use the Pythagorean Theorem.
a2 b2 c2
(2.1 km)2 (1.7 km)2 (length of marsh)2
4.41 km2 2.89 km2 (length of marsh)2
7.3 km2 (length of marsh)2
2.7 km length of marsh
When solving word
problems, be sure you
have correctly identified
the hypotenuse. It will
always be opposite the
right angle.
Quick Check 2
2a. Mr. Jones measures a marsh by starting at
point A and driving straight toward one end
of the marsh, covering 1.2 km. Driving from
point A to the other end of the marsh covers
2.8 km. How long is the marsh, to the
nearest tenth of a kilometer?
2b. On a hike, you follow a river running due
south for about 1.5 miles. Turning due west,
you walk another 0.8 mile to a meadow.
How many miles is it from the meadow to
your starting point?
A
1.2 km
2.8 km
marsh
TAKS Review and Preparation Workbook
LESSON 18
■
The Pythagorean Theorem
53
Name__________________________Class____________Date________
1 Which answer is closest to the length of line
segment d in the following diagram?
4 The diagram shows an equilateral triangle
bisected by a dotted line.
7
5
d
3
6
11
d
6
6
A 7.6
C 9.6
B 8.1
D 9.8
Which of the following best represents the
height, d, of the triangle?
F 5.2
G 6.0
2 An archaeologist measures a square
pyramid. The length of one side of the base
of the pyramid is 600 m. The length of each
face of the pyramid is 500 m. Which of the
following is closest to the height of the
pyramid?
H 6.7
J 8.5
5 Which drawing gives you enough information
to find the length of line segment d?
500 m
600 m
A
7
7
G 300 m
d
H 400 m
12
J 800 m
B
6
3 The base of a 30-foot ramp is placed 26 feet
away from a building. About how high from
the ground does the ramp reach?
d
d
C
3
5
ramp
30 ft
26 ft
54
D
A 7.5 feet
C 26 feet
B 15 feet
D 40 feet
LESSON 18
■
The Pythagorean Theorem
4
d
2
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
F 200 m