1. 4-9 The Pythagorean Theorem Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview

2. 4-9 The Pythagorean Theorem 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. 4-9 The Pythagorean Theorem 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. 4-9 The Pythagorean Theorem Pythagorean Theorem leg hypotenuse Vocabulary

5. 4-9 The Pythagorean Theorem 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. 4-9 The Pythagorean Theorem Use the Pythagorean Theorem to find the missing measure. Additional Example 1A: Calculating the Length of a Side of a Right Triangle 12 cm 16 cm a 2 + b 2 = c 2 c 12 2 + 16 2 = c 2 144 + 256 = c 2 400 = c 2 The length of the hypotenuse is 20 cm. Use the Pythagorean Theorem. Substitute for a and b. Evaluate the powers. Add. Take the square root of both sides. 20 = c √ 400 = √ c 2

7. 4-9 The Pythagorean Theorem Additional Example 1B: Calculating the Length of a Missing Side of a Right Triangle Use the Pythagorean Theorem to find the missing measure. 5 cm b a 2 + b 2 = c 2 13 cm 5 2 + b 2 = 13 2 25 + b 2 = 169 b 2 = 144 The length of the missing leg is 12 cm. Use the Pythagorean Theorem. Substitute for a and c. Evaluate the powers. Take the square root of both sides. b = 12 –25 Subtract 25 from each side. √ b 2 = √ 144

8. 4-9 The Pythagorean Theorem Use the Pythagorean Theorem to find the missing measure. 11 cm 15 cm a 2 + b 2 = c 2 c 11 2 + 15 2 = c 2 121 + 225 = c 2 346 = c 2 The length of the hypotenuse is about 18.6 cm. Use the Pythagorean Theorem. Substitute for a and b. Evaluate the powers. Add. Take the square root of both sides. 18.6  c Check It Out! Example 1A √ 346 = √ c 2

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

10. 4-9 The Pythagorean Theorem 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.

11. 4-9 The Pythagorean Theorem Additional Example 2 Continued The segment between the two corners is the hypotenuse. The sides of the field are legs, and they are each 75 feet long. 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. 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.

12. 4-9 The Pythagorean Theorem Additional Example 2 Continued 2 Make a Plan You can use the Pythagorean Theorem to write an equation.

13. 4-9 The Pythagorean Theorem Additional Example 2 Continued Solve 3 a 2 + b 2 = c 2 75 2 + 75 2 = c 2 5,625 + 5,625 = c 2 11,250 = c 2 106.066  c The distance from one corner of the field to the opposite corner is about 106.1 feet. Use the Pythagorean Theorem. Substitute for the known variables. Evaluate the powers. Add. Take the square roots of both sides. 106.1  c Round.

14. 4-9 The Pythagorean Theorem Additional Example 2 Continued Look Back 4 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.

15. 4-9 The Pythagorean Theorem 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.

16. 4-9 The Pythagorean Theorem Check It Out! 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. 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. 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.

17. 4-9 The Pythagorean Theorem Check It Out! Example 2 Continued 2 Make a Plan You can use the Pythagorean Theorem to write an equation.

18. 4-9 The Pythagorean Theorem Check It Out! Example 2 Continued Solve 3 a 2 + b 2 = c 2 33 2 + 100 2 = c 2 1089 + 10,000 = c 2 11,089 = c 2 105.304  c The distance from one corner of the field to the opposite corner is about 105.3 yards. Use the Pythagorean Theorem. Substitute for the known variables. Evaluate the powers. Add. Take the square roots of both sides. 105.3  c Round.

19. 4-9 The Pythagorean Theorem Check It Out! Example 2 Continued Look Back 4 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 either side of the field, the answer is reasonable.

20. 4-9 The Pythagorean Theorem Tell whether the given side lengths form a right triangle. Additional Example 3A: Identifying a Right Triangle a 2 + b 2 = c 2 12 2 + 35 2 = 37 2 144 + 1225 = 1369 The side lengths form a right triangle. Compare a 2 to b 2 to c 2. Substitute the longest side length for c. Simplify the powers. Add.1369 = 1369 A. 12, 35, 37 ? ? ?

21. 4-9 The Pythagorean Theorem Tell whether the given side lengths form a right triangle. Additional Example 3B: Identifying a Right Triangle a 2 + b 2 = c 2 8 2 + 12 2 = 16 2 64 + 144 = 256 The side lengths do not form a right triangle. Compare a 2 to b 2 to c 2. Substitute the longest side length for c. Simplify the powers. Add.208 ≠ 256 B. 8, 12, 16 ? ? ? 

22. 4-9 The Pythagorean Theorem Tell whether the given side lengths form a right triangle. Check It Out! Example 3A a 2 + b 2 = c 2 10 2 + 15 2 = 20 2 100 + 225 = 400 The side lengths do not form a right triangle. Compare a 2 to b 2 to c 2. Substitute the longest side length for c. Simplify the powers. Add.325 ≠ 400 A. 10, 15, 20 ? ? ? 

23. 4-9 The Pythagorean Theorem Tell whether the given side lengths form a right triangle. Check It Out! Example 3B a 2 + b 2 = c 2 8 2 + 15 2 = 17 2 64 + 225 = 289 The side lengths form a right triangle. Compare a 2 to b 2 to c 2. Substitute the longest side length for c. Simplify the powers. Add.289 = 289 B. 8, 15, 17 ? ? ?

24. 4-9 The Pythagorean Theorem Lesson Quiz 21 in. 40 m Use the Pythagorean Theorem to find each missing measure. 1. 2. 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 no yes 5. 33, 56, 65 4. 2.5, 3, 4.5 Tell whether the given side lengths form a right triangle.