Pythagorean Theorem
c2
a=____ b=____ c=____
a2
b2
In any right triangle, the sum of
the square of the lengths of the
legs is equal to the square length
of the hypotenuse
1
LESSON
11.2 The Pythagorean Theorem
Bridges The William H.
Harsha Bridge is a cablestayed bridge that spans the
Ohio River between
Maysville, Kentucky, and
Aberdeen, Ohio. About how
long is the cable shown in
red?
2
LESSON
11.2 The Pythagorean Theorem
In a right triangle, the hypotenuse is the side opposite the right angle.
The legs are the sides that form the right angle. The lengths of the legs
and the length of the hypotenuse of a right triangle are related by the
Pythagorean theorem.
3
LESSON
11.2 The Pythagorean Theorem
Pythagorean Theorem
Words
For any right triangle, the sum of the squares of the
lengths of the legs equals the square of the length
of the hypotenuse.
Algebra
a2 + b2 = c2
4
LESSON
11.2 The Pythagorean Theorem
EXAMPLE
1
Finding the Length of a Hypotenuse
To find the length (to the nearest foot) of the cable on
the William H. Harsha bridge if the tower is 212 feet and
bridge surface is 478 feet, use the right triangle formed
by the tower, the bridge surface, and the cable.
a2 + b2 = c2
212 2 + 478 2 = c 2
44,944 + 228,484 = c 2
Pythagorean theorem
Substitute 212 for a and 478 for b.
Evaluate powers.
273,428 = c 2
Add.
273,428 = c
Take positive square root of each side.
523
ANSWER
c
Use a calculator. Round to nearest whole number.
The length of the cable is about 523 feet.
5
LESSON
11.2 The Pythagorean Theorem
EXAMPLE
2
Finding the Length of a Leg
Find the unknown length a in simplest form.
a2 + b2 = c2
Pythagorean theorem
a 2 + 10 2 = 12 2
Substitute.
a 2 + 100 = 144
Evaluate powers.
a 2 = 44
ANSWER
Subtract 100 from each side.
a = 44
Take positive square root of each side.
a = 2 11
Simplify.
The unknown length a is 2 11 units.
6
LESSON
11.2 The Pythagorean Theorem
Converse of the Pythagorean Theorem The Pythagorean theorem can be
written in “if-then” form.
Theorem: If a triangle is a right triangle, then a 2 + b 2 = c 2.
If you reverse the two parts of the statement, the new statement is called
the converse of the Pythagorean theorem.
Converse: If a 2 + b 2 = c 2, then the triangle is a right triangle.
Although not all converses of true statements are true, the converse of
the Pythagorean theorem is true. You can use it to determine whether
a triangle is a right triangle.
7
LESSON
11.2 The Pythagorean Theorem
EXAMPLE
3
Identifying Right Triangles
Determine whether the triangle with the given side lengths is a right triangle.
a = 3, b = 5, c = 7
SOLUTION
a2 + b2 = c2
?
32 + 52 = 72
?
9 + 25 = 49
34 = 49
ANSWER
Not a right triangle.
8
LESSON
11.2 The Pythagorean Theorem
EXAMPLE
3
Identifying Right Triangles
Determine whether the triangle with the given side lengths is a right triangle.
a = 3, b = 5, c = 7
SOLUTION
a = 15, b = 8, c = 17
SOLUTION
a2 + b2 = c2
a2 + b2 = c2
?
15 2 + 8 2 = 17 2
9 + 25 = 49
?
225 + 64 = 289
34 = 49
289 = 289
32 + 52 = 72
ANSWER
Not a right triangle.
?
?
ANSWER
A right triangle.
9
Lesson 11-2
The Pythagorean Theorem
Additional Examples
Find c, the length of the hypotenuse.
c2 = a2 + b2
Use the Pythagorean Theorem.
c2 = 282 + 212
Replace a with 28, and b with 21.
c2 = 1,225
Simplify.
c = 1,225 = 35 Find the positive square root of each side.
The length of the hypotenuse is 35 cm.
Quick Check
10
11-2
Lesson 11-2
The Pythagorean Theorem
Additional Examples
Find the value of x in the triangle.
Round to the nearest tenth.
a2 + b2 = c2
Use the Pythagorean Theorem.
72 + x2 = 142
Replace a with 7, b with x, and c with 14.
49 + x2 = 196
Simplify.
x2 = 147
Subtract 49 from each side.
x = 147 Find the positive square root of each side.
11
11-2
Lesson 11-2
The Pythagorean Theorem
Additional Examples
(continued)
Then use one of the two methods below to
approximate 147 .
Method 1: Use a calculator.
A calculator value for
147 is 12.124356.
x 12.1 Round to the nearest tenth.
Method 2: Use a table of square roots.
Use the table on page 800. Find the number
closest to 147 in the N2 column. Then find the
corresponding value in the N column. It is a
little over 12.
x 12.1
Estimate the nearest tenth.
The value of x is about 12.1 in.
Quick Check
12
11-2
Lesson 11-2
The carpentry terms span, rise, and
rafter length are illustrated in the diagram. A
carpenter wants to make a roof that has a span
of 20 ft and a rise of 10 ft. What should the
rafter length be?
c2 = a2 + b2
Use the Pythagorean Theorem.
c2 = 102 + 102 Replace a with 10 (half the span), and b with 10.
c2 = 100 + 100 Square 10.
c2 = 200
Add.
c = 200 Find the positive square root.
c 14.1
Round to the nearest tenth.
Quick Check
The rafter length should be about 14.1 ft.
13
11-2
Lesson 11-2
The Pythagorean Theorem
Is a triangle with sides 10 cm, 24 cm,
and 26 cm a right triangle?
2
a2 + b 2 = c
Write the equation for the Pythagorean
Theorem.
102 + 242 262 Replace a and b with the shorter
lengths and c with the longest
length.
100 + 576
676 Simplify.
676 = 676
The triangle is a right triangle.
Quick Check
14
11-2