2. REVIEW OF ROOTS 4 This is a collection of warm-ups and practice from class. 4 Click to advance the slide and follow along. 4 You can use the scroll bar at the right to “fast forward” or rewind the slides.

3. WARM-UP 4 What is the prime factorization of the following numbers (ex. 14=2 7): 9, 16, 25, 10, 17, 24, 27 4 What are the square roots of the following numbers: 9, 16, 25, 10, 17, 24, 27

4. Answers 4 9 = 33 4 16 = 2222 4 25 = 55 4 10 = 25 4 17 = 17 4 24 = 2223 4 27 = 333  9 = 3  16 = 4  25 = 5  10 =  10 or  3.2  17 =  17 or  4.1  24 = 2  6 or  4.9  27 = 3  3 or  5.2

5. Chapter 10 – Right Triangles 4 Why should you care? 4 LOTS OF STANDARDS: Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. 4 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. 4 Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.

6. Chapter 10 – Right Triangles 4 Why should you care? 4 TRIGONOMETRY: You’ll need this stuff for next year!! 4 Mr. Taylor’s Opinion. Pythagorean theorem and the trigonometric functions are EXTREMELY useful for DOING practical problems involving graphing(drawing) and measurement.

7. ROOTS – From Chapter 3 4 WHY NOT USE A CALCULATOR? –What is the square root of 5 on a calculator? – approximately 2.236 –What’s the square root of 5 squared? –5–5 –What’s the 2.236 squared? –4.999696 (close to 5 but not exactly) 4 THEREFORE, if we want exactly the square root of 5, use

8. Finding exact roots 4 To simplify a number which includes a radical, find the prime factorization of the radicand and move all the perfect squares out front. 4 Examples. 4, 6, 8, 12, 15, 18

9. Pythagorean Theorem 4 The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a 2 + b 2 = c 2 1. If a right triangle has legs 3 and 4, what is the length of the hypotenuse? 2. If a right triangle has a leg 2 and a hypotenuse  10, what is the length of the other leg? 3. If a triangle has sides 5, 6, and 8 is it a right triangle?

10. Pythagorean Theorem 4 The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a 2 + b 2 = c 2 1. If a right triangle has legs 3 and 4, what is the length of the hypotenuse? 3 2 + 4 2 = c 2 9 + 16 = c 2 25 = c 2  25 =  c 2 5 = c

11. Pythagorean Theorem 4 The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a 2 + b 2 = c 2 2. If a right triangle has a leg 2 and a hypotenuse  10, what is the length of the other leg? 2 2 + b 2 = (  10 ) 2 4 + b 2 = 10 b 2 = 10 – 4 b 2 = 6  b 2 =  6 b =  6

12. Pythagorean Theorem 4 The Pythagorean Theorem states that for any triangles with legs a and b and hypotenuse c that a 2 + b 2 = c 2 2. If a triangle has sides 5, 6, and 8 is it a right triangle? 5 2 + 6 2 = 8 2 25 + 36 = 64 51 not equal 64 NO, Not a right triangle

13. WARM-UP 4 What are the lengths of the missing sides? 45 90 45 7 7 7272 30 90 60 12 6 6363 6 2 + b 2 = 12 2 b 2 = 144 – 36 b =  108 b = 6  3 7 2 + 7 2 = c 2 c 2 = 49+49 c =  2(7)(7) c = 7  2

14. Gold Boxes from p. 424 4 In a 45-45-90 Triangle, the measure of the hypotenuse is  2 times the leg. 45 90 45 x x x2x2 x 2 + x 2 = c 2 c 2 = x 2 + x 2 c 2 = 2x 2 c =  2 x 2 c = x  2

15. 45-45-90 is an isosceles triangle 4 What about this? 45 90 45 7272 7272 7  2  2 = 7 * 2 = 14

16. 45-45-90 is an isosceles triangle 45 90 45 7676 7676 7  6  2 = 7  2*2*3 = 7 * 2 *  3 = 14  3

17. 45-45-90 is an isosceles triangle 45 90 45 3 3 3232

18. 45-45-90 is an isosceles triangle 45 90 45 Better  7 This answer Would never Be on a M.C. Test

19. 45-45-90 is an isosceles triangle 45 90 45 Better  8

20. Preview 30/60/90

21. 30-60-90 is half an equilateral triangle 4 30-60-90 (Assume short side is opposite small angle) 30 90 60 8 4 4343 4 2 + X 2 = 8 2 X 2 = 64 – 16 X =  48 X = 4  3 30/60/90 is half of an equilateral (60/60/60) triangle. Therefore, the side opposite the 30 will always be half of the side opposite the 90 and the side opposite the 90 will always be twice the side opposite the 30.

22. 30-60-90 is half an equilateral triangle 30 90 60 14 7 7373 7 2 + X 2 = 14 2 X 2 = 196 – 49 X =  147 X = 7  3

23. 30-60-90 is half an equilateral triangle 30 90 60 12 6 6363 6 2 + X 2 = 12 2 X 2 = 144 – 36 X =  108 X = 6  3

24. 30-60-90 is half an equilateral triangle 30 90 60 2m m m3m3 m 2 + X 2 = (2m) 2 X 2 = 4m 2 – m 2 X 2 = 3m 2 X = m  3

25. WARM-UP 4 What are the lengths of the missing sides? 45 90 45 9 9 9292 30 90 60 10 5 5353 5 2 + b 2 = 10 2 b 2 = 100 – 25 b =  75 b = 5  3 9 2 + 9 2 = c 2 c 2 = 81+81 c =  2(9)(9) c = 9  2

26. 45-45-90 is an isosceles triangle 45 90 45 9 9 9292

27. 45-45-90 is an isosceles triangle 45 90 45 5 5 5252

28. 45-45-90 is an isosceles triangle 45 90 45 6262 6262 6  2  2 = 6 * 2 = 12

29. 45-45-90 is an isosceles triangle 45 90 45 7676 7676 7  6  2 = 7 * 2 *  3 = 14  3

30. 45-45-90 is an isosceles triangle 45 90 45 3 3 3232

31. 45-45-90 is an isosceles triangle 45 90 45 Better  7 This answer Would never Be on a M.C. Test

32. 45-45-90 is an isosceles triangle 45 90 45 Better  10

33. 30-60-90 is half an equilateral triangle 4 30-60-90 (Assume short side is opposite small angle) 30 90 60 8 4 4343 4 2 + X 2 = 8 2 X 2 = 64 – 16 X =  48 X = 4  3

34. 30-60-90 is half an equilateral triangle 30 90 60 14 7 7373 7 2 + X 2 = 14 2 X 2 = 196 – 49 X =  147 X = 7  3

35. 30-60-90 is half an equilateral triangle 30 90 60 2m m m3m3 m 2 + X 2 = (2m) 2 X 2 = 4m 2 – m 2 X 2 = 3m 2 X = m  3

36. Rules: 4 In a 45-45-90 Triangle, the measure of the hypotenuse is the leg times  2 4 In a 30-60-90 Triangle: hypotenuse = 2 x shorter leg longer leg =  3 x shorter leg 4 To go in reverse direction, reverse the operation. –For instance, to go from hypotenuse to leg in a 45-45- 90, divide by  2 4 Answers must have no perfect squares under the radicals and no radicals in the denominator.

37. 30-60-90 is half an equilateral triangle 30 90 60 426 213 213  3

38. 30-60-90 is half an equilateral triangle 30 90 60 2 1 33

39. 30-60-90 is half an equilateral triangle 30 90 60 4 2 2323

40. 30-60-90 is half an equilateral triangle 30 90 60 5

41. 30-60-90 is half an equilateral triangle 30 90 60 6