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
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 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. 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 – 25 Subtract 25 from each side. √ b 2 = √ 144
8. 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. 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 – 9 Subtract 9 from each side. Check It Out! Example 1B √ b 2 = √ 16
10. 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. 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. Rewrite the question as a statement. • Find the distance from one corner of the field to the opposite corner of the field. 1 Understand the Problem
12. Additional Example 2 Continued You can use the Pythagorean Theorem to write an equation. 2 Make a Plan
13. Additional Example 2 Continued 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. Solve 3
14. 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. 4
15. 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. Check It Out! 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. • 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. 1 Understand the Problem
17. Check It Out! Example 2 Continued You can use the Pythagorean Theorem to write an equation. 2 Make a Plan
18. Check It Out! Example 2 Continued 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. Solve 3
19. Check It Out! 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 either side of the field, the answer is reasonable. 4
20. 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. 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. 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. 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. 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.
