1. Preview
Warm Up
California Standards
Lesson Presentation

2. Warm Up
Estimate each square root to nearest hundredth.
1. √30
2. √14
3. √55
4. √48
5.48
3.74
7.42
6.93

3. MG3.3 Know and understand the Pythagorean theorem and its converse and use it find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
California
Standards

4. Vocabulary
Pythagorean Theorem
leg
hypotenuse

5. The Pythagorean Theorem shows that a special relationship exists between the sides of a right triangle.
You can use the theorem to find the length of any side of a right triangle.

6. Use the Pythagorean Theorem to find the missing measure.
Additional Example 1A: Calculating the Length of a Side of a Right Triangle
Use the Pythagorean Theorem to find the missing measure. c
12 cm
16 cm
a2 + b2 = c2
Use the Pythagorean Theorem.
= c2
Substitute for a and b.
= c2
Evaluate the powers.
400 = c2
Add.
√400 = √c2
Take the square root of both sides.
20 = c
The length of the hypotenuse is 20 cm.

7. Use the Pythagorean Theorem to find the missing measure.
Additional Example 1B: Calculating the Length of a Missing Side of a Right Triangle
Use the Pythagorean Theorem to find the missing measure. b
5 cm
13 cm
a2 + b2 = c2
Use the Pythagorean Theorem.
52 + b2 = 132
Substitute for a and c.
25 + b2 = 169
Evaluate the powers.
–25
–25
Subtract 25 from each side.
b2 = 144
√b2 = √144
Take the square root of both sides.
b = 12
The length of the missing leg is 12 cm.

8. Additional Example 2: Problem Solving Application
A square field has sides of 75 feet. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth.
75ft
75ft
75ft
75ft

9. Understand the Problem
Additional Example 2 Continued 1
Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of the field
to the opposite corner of the field.
List the important information:
• Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles.
• The segment between the two corners is
the hypotenuse.
• The sides of the field are legs, and they
are each 75 feet long.

10. Additional Example 2 Continued
Make a Plan
You can use the Pythagorean Theorem to
write an equation.

11. Additional Example 2 Continued
Solve 3
a2 + b2 = c2
Use the Pythagorean Theorem.
= c2
Substitute for the known variables.
5, ,625 = c2
Evaluate the powers.
11,250 = c2
Add.
 c
Take the square roots of both sides.
106.1  c
Round.
The distance from one corner of the field to the
opposite corner is about feet.

12. Additional Example 2 Continued4
Look Back
The hypotenuse is the longest side of a right triangle. Since the distance from one corner of the field to the opposite corner is greater than the length of a side of the field, the answer is reasonable.

13. Additional Example 3A: Identifying a Right Triangle
Tell whether the given side lengths form a right triangle.
A. 12, 35, 37 ?
a2 + b2 = c2
Compare a2 to b2 to c2. ?
= 372
Substitute the longest side length for c. ?
= 1369
Simplify the powers.
1369 = 1369 
Add.
The side lengths form a right triangle.

14. Additional Example 3B: Identifying a Right Triangle
Tell whether the given side lengths form a right triangle.
B. 8, 12, 16 ?
a2 + b2 = c2
Compare a2 to b2 to c2. ?
= 162
Substitute the longest side length for c. ?
= 256
Simplify the powers.
208 ≠ 256 
Add.
The side lengths do not form a right triangle.

15. Lesson Quiz
Use the Pythagorean Theorem to find each missing measure. 1. 2.
21 in.
40 m
3. Each rectangular section of a fence is braced by a board nailed on the diagonal of the section. The fence is 6 ft tall and the brace is 10 ft long. What is the length of the section?
8 ft
Tell whether the given side lengths form a right triangle.
4. 2.5, 3, 4.5 no
5. 33, 56, 65
yes