1. Exercise Solve x 2 = 4. x = ± 2

2. Solve x 2 = – 4. no real solution Exercise

3. Solve √ x = 4. x = 16 Exercise

4. Solve √ x = – 4. no real solution Exercise

5. Solve √ – x = 4. x = – 16 Exercise

6. leg hypotenuse

7. 55 33 44 9 square units 25 square units 16 square units

8. 55 33 44 9 square units 25 square units 16 square units + =

9. 16 9 25 4 2 3 2 5 2

10. The Pythagorean Theorem If the hypotenuse of a right triangle has length c, and the legs have lengths a and b, then a 2 + b 2 = c 2.

11. Find the hypotenuse of a right triangle with legs of 8 and 15. c = 17 c 2 = a 2 + b 2 c 2 = 8 2 + 15 2 c 2 = 64 + 225 c 2 = 289 √ c 2 = √ 289 Example 1

12. Find the hypotenuse of a right triangle with legs of 6 and 7. c 2 = a 2 + b 2 c 2 = 6 2 + 7 2 c 2 = 36 + 49 c 2 = 85 √ c 2 = √ 85 c = √ 85 ≈ 9.2 Example 2

13. Find the hypotenuse of a right triangle with legs of 9 and 12. 15 Example

14. Find the hypotenuse of a right triangle with legs of and. 11 1212 1212 √ 32√ 32 √ 32√ 32 Example

15. Find the hypotenuse of a right triangle with legs of 1 and 1. √ 2√ 2√ 2√ 2 Example

16. Find the leg of a right triangle whose hypotenuse is 16 and other leg is 7. a 2 + 7 2 = 16 2 a 2 + 49 = 256 a 2 = 207 a = √ 207 ≈ 14.4 a 2 + 49 – 49 = 256 – 49 Example 3

17. Find the length of a leg of a right triangle whose hypotenuse is 39 and whose other leg is 15. 36 Example

18. Find the length of a leg of a right triangle whose hypotenuse is 20 and whose other leg is 10. √ 300 ≈ 17.3 Example

19. The converse is the statement resulting when the “if” part and the “then” part of a conditional statement are switched. Converse

20. Converse of the Pythagorean Theorem If a triangle has sides a, b, and c, such that a 2 + b 2 = c 2, then the triangle is a right triangle.

21. Determine whether a triangle with sides of 12, 35, and 37 is a right triangle. 1,369 = 1,369 a 2 + b 2 = c 2 12 2 + 35 2 = 37 2 144 + 1,225 = 1,369 Example 4 yes

22. Determine whether a triangle with sides of 8, 12, and 14 is a right triangle. 208 ≠ 196 a 2 + b 2 = c 2 8 2 + 12 2 = 14 2 64 + 144 = 196 Example 5 no

23. Determine whether a triangle with sides of 15, 18, and 22 is a right triangle. no; 15 2 + 18 2 ≠ 22 2 Example

24. Determine whether a triangle with sides of 16, 30, and 34 is a right triangle. yes; 16 2 + 30 2 = 1,156 = 34 2 Example

25. A 16 ft. ladder leans up against the side of a building. If the base of the ladder is 4 ft. from the base of the building, how high up the side of the building will the ladder reach? 15.5 ft. Exercise

26. A 200 ft. tower is braced to the ground by a cable, from a point 150 ft. above the ground to a point 87 ft. from the base of the tower. How long is the cable? 173.4 ft. Exercise

27. The distance between bases on a baseball diamond is 90 ft. How far is it from home plate to second base? 127.3 ft. Exercise

28. An opening for a window is 23” wide, 54” tall, and 60” diagonally. Is the opening “square”; that is, do the height and width form a right angle? no Exercise