Learn to use the Pythagorean Theorem to find the length of a side of a right triangle.

1. 9-8 The Pythagorean Theorem Course 2 Learn to use the Pythagorean Theorem to find the length of a side of a right triangle.

2. 9-8 The Pythagorean Theorem Course 2 Insert Lesson Title Here Vocabulary leg hypotenuse Pythagorean Theorem

3. 9-8 The Pythagorean Theorem Course 2 In a right triangle, the two sides that form the right angle are called legs. The side opposite the right angle is called the hypotenuse. Hypotenuse Leg Leg One of the first people to recognize the relationship between the sides of a right triangle was the Greek mathematician Pythagoras. This special relationship is called the Pythagorean Theorem.

4. 9-8 The Pythagorean Theorem Course 2 PYTHAGOREAN THEOREM In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. c a a2 + b2 = c2 b You can use the Pythagorean Theorem to find the length of any side of a right triangle.

5. 9-8 The Pythagorean Theorem √400 = √c2 Course 2 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 Use the Pythagorean Theorem. a2 + b2 = c2 Substitute for a and b. 122+ 162 = c2 Evaluate the powers. 144 + 256 = c2 Add. 400 = c2 Take the square root of both sides. 20 = c The length of the hypotenuse is 20 cm.

6. 9-8 The Pythagorean Theorem √b2 = √144 Course 2 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 Use the Pythagorean Theorem. a2 + b2 = c2 Substitute for a and c. 52+ b2 = 132 25 + b2 = 169 Evaluate the powers. –25 Subtract 25 from each side. –25 b2= 144 Take the square root of both sides. b = 12 The length of the missing leg is 12 cm.

7. 9-8 The Pythagorean Theorem √346 = √c2 Course 2 Check It Out: Example 1A Use the Pythagorean Theorem to find the missing measure. c 11 cm 15 cm Use the Pythagorean Theorem. a2 + b2 = c2 Substitute for a and b. 112+ 152 = c2 Evaluate the powers. 121 + 225 = c2 Add. 346 = c2 Take the square root of both sides. 18.6 c The length of the hypotenuse is about 18.6 cm.

8. 9-8 The Pythagorean Theorem √b2 = √ 16 Course 2 Check It Out: Example 1B Use the Pythagorean Theorem to find the missing measure. b 3 cm 5 cm Use the Pythagorean Theorem. a2 + b2 = c2 Substitute for a and c. 32+ b2 = 52 9 + b2 = 25 Evaluate the powers. –9 Subtract 9 from each side. –9 b2= 16 Take the square root of both sides. b = 4 The length of the missing leg is 4 cm.

9. 9-8 The Pythagorean Theorem Course 2 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.

10. 9-8 The Pythagorean Theorem 1 Understand the Problem Course 2 Additional Example 2 Continued 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.

11. 9-8 The Pythagorean Theorem Make a Plan 2 Course 2 Additional Example 2 Continued You can use the Pythagorean Theorem to write an equation.

12. 9-8 The Pythagorean Theorem 3 Solve Course 2 Additional Example 2 Continued a2+ b2 = c2 Use the Pythagorean Theorem. Substitute for the known variables. 752 + 752 = c2 5,625 + 5,625 = c2 Evaluate the powers. 11,250 = c2 Add. Take the square roots of both sides. 106.066012 c Round. 106.1 c The distance from one corner of the field to the opposite corner is about 106.1 feet

13. 9-8 The Pythagorean Theorem 4 Course 2 Additional Example 2 Continued 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.

14. 9-8 The Pythagorean Theorem 1 Understand the Problem Course 2 Insert Lesson Title Here Check It Out: Example 2 A rectangular field has a length of 100 yards and a width of 33 yards. 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. Rewrite the question as a statement. • Find the distance from one corner of the field to the opposite corner of the field.

15. 9-8 The Pythagorean Theorem Make a Plan 2 Course 2 Check It Out: Example 2 Continued 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 fields are legs, and they are 33 yards long and 100 yards long. You can use the Pythagorean Theorem to write an equation.

16. 9-8 The Pythagorean Theorem 3 Solve Course 2 Insert Lesson Title Here Check It Out: Example 2 Continued a2+ b2 = c2 Use the Pythagorean Theorem. 332 + 1002 = c2 Substitute for the known variables. 1089 + 10,000 = c2 Evaluate the powers. 11,089 = c2 Add. 105.3043208 c Take the square roots of both sides. 105.3 c Round. The distance from one corner of the field to the opposite corner is about 105.3 yards.

17. Facts about Sides and angles a2+ b2 > c2 a2+ b2 = c2 How to determine if a triangle is a right triangle. The largest square is less than the smaller squares combined, . Acute triangle The largest square is equal to the smaller squares combined, . Right triangle The largest square is greater than the smaller square combined, . Obtuse triangle a2+ b2 < c2