Lesson 9.1 Using Similar Right Triangles

2. Lesson 9.1Using Similar Right Triangles Students need scissors, rulers, and note cards. Today, we are going to… …use geometric mean to solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle

3. With a straight edge, draw one diagonal of the note card. Draw an altitude from one vertex of the note card to the diagonal. Cut the note card into three triangles by cutting along the segments.

4. C C B D A B D A B hyp hypotenuse long hyp long leg long leg short short leg short leg Color code all 3 sides of all 3 triangles on the front and back.

5. Arrange the small and medium triangles on top of the large triangle like this.?

6. Theorems 9.1 – 9.3 If the altitude is drawn to the hypotenuse of a right triangle, then…

7. Theorem 9.1…the three triangles formed are similar to each other.

8. B C A D AD BD BD is a side of the medium  and a side of the small  = BD CD

9. BD AD BD CD x n m = = x n m x

10. B x n m x A C = D Theorem9.2…the altitude is the geometric mean of the two segments of the hypotenuse. x m n

11. When you do these problems, always tell yourself… ____ is the geometric mean of ____ and ____

12. x 3 8 = 1. Find x. x 3 x 8 x ≈ 4.9

13. 4 x 8 = 2. Find x. 4 x 4 8 x = 2

14. B C A D CB CA CB is a side of the large  and a side of the medium  = CD CB

15. CB CA CB CD x m = = h x h m x

16. B C A D AB AC AB is a side of the large  and a side of the small  = AD AB

17. AB AC AB AD x n h = = x h n x

18. B y x h h m n y x A C = = D Theorem 9.3…the leg of the large triangle is the geometric mean of the “adjacent leg” and the hypotenuse. x y m n h

19. When you do these problems, always tell yourself… ____ is the geometric mean of ____ and ____

20. x 9 = 14 3. Find x. x 14 x 9 x ≈ 11.2

21. x 4 10 = 4. Find x. x 10 x 4 x ≈ 6.3

22. = = = 5. Find x, y, z x z x y 13 x 9 x ≈ 10.8 9 4 y 4 z 13 y 9 z 4 y = 6 z ≈ 7.2

23. 3 5 = = 5 6. Find h. 3 x x = 1.8 3.2 h 1.8 4 h 3.2 x 1.8 h h = 2.4 3

25. Lesson 9.2 & 9.3 The Pythagorean Theorem & Converse Today, we are going to… …prove the Pythagorean Theorem …use the Pythagorean Theorem and its Converse to solve problems

26. Theorem 9.4Pythagorean Theorem hyp2 = leg2 + leg2(c2 = a2 + b2 )

27. 102 = y2 + 82 2. Find x. 100 = y2 + 64 Why can’t we use a geo mean proportion? 36 = y2 y = 6 x 10 8 x2 = 142 + 82 x2 = 196 + 64 y 14 x2 = 260 20 x ≈ 16.1

28. A Pythagorean Triple is a set of three positive integers that satisfy the equation c 2 = a 2 + b 2. The integers 3, 4, and 5 form a Pythagorean Triple because 52 = 32 + 42.

29. Theorem 9.5Converse of the Pythagorean TheoremIf c2 = a2 + b2, then the triangle is a right triangle.

30. The hypotenuse is the perfect length for the opposite angle to be 90˚ c2 = a2 + b2 The “hypotenuse” is too long for the opposite angle to be 90˚ c2 > a2 + b2 The “hypotenuse” is too short for the opposite angle to be 90˚ c2 < a2 + b2

31. Theorem 9.6If c2<a2 + b2, then the triangle is an acute triangle.

32. Theorem 9.7If c2> a2 + b2, then the triangle is an obtuse triangle.

33. How do we know if 3 lengths can represent the side lengths of a triangle? L < M + S

34. What kind of triangle? Can a triangle be formed? L2 = M2 + S2 L = M + S Right  No  can be formed L2 < M2 + S2 L < M + S Acute  Yes,  can be formed L2 > M2 + S2 L > M + S No  can be formed Obtuse 

35. Do the lengths represent the lengths of a triangle? Is it a right triangle, acute triangle, or obtuse triangle? 3. 10, 24, 26 262 = 102 + 242 right triangle 26 < 10 + 24? 72 > 32 + 52 4. 3, 5, 7 obtuse triangle 7 < 5 + 3? 5. 5, 8, 9 92 < 52 + 82 9 < 8 + 5? acute triangle

36. 92 = 52 + 56 2 9 < 5 + 56 ? Do the lengths represent the lengths of a triangle? Is it a right triangle, acute triangle, or obtuse triangle? 6. 5, 56 , 9 right triangle 7. 23, 44, 70 not a triangle 70 < 23 + 44? 8. 12, 80, 87 872 > 122 + 802 87 < 12 + 80? obtuse triangle

37. Find the area of the triangle. 9. 92 = x2 + 62 6 81 = x2 + 36 9 45 = x2 A ≈ ½ (6)(6.7) 6.7 = x ≈ 20.12 units2

38. Find the area of the triangle. 10. 13 202 = x2 + 132 400 = x2 + 169 20 231 = x2 15.2 = x A ≈ ½ (13)(15.2) ≈ 98.8 units2

39. 7 2 Find the area of the triangle. 11. 72 = x2 + 22 x 49 = x2 + 4 45 = x2 6.7 = x A ≈ ½ (4)(6.7) ≈ 13.4 units2

40. 302 + 182 = Which method requires less ribbon? How much ribbon is needed using Method 1? (3+12+3+12) + (3+6+3+6) = 48 in. The diagram shows the ribbon for Method 2. How much ribbon is needed to wrap the box? 30 ? 18 ? 35 in

41. Project ideas… A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. How far up on the building will the ladder reach? A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. How long is the cable? The library is 5 miles north of the bank. Your house is 7 miles west of the bank. Find the distance from your house to the library.

42. Project ideas… Playing baseball, the catcher must throw a ball to 2nd base so that the 2nd base player can tag the runner out. If there are 90 feet between home plate and 1st base and between 1st and 2nd bases, how far must the catcher throw the ball? While flying a kite, you use 100 feet of string. You are standing 60 feet from the point on the ground directly below the kite. Find the height of the kite.

44. Lesson 9.4Special Right Triangles Today, we are going to… …find the side lengths of special right triangles

45. 45˚ y x 45˚ 5 1. Find x. Use the Pythagorean Theorem to find y. Leave y in simplest radical form. x = 5

46. 45˚ y x 45˚ 9 2. Find x. Use the Pythagorean Theorem to find y. Leave y in simplest radical form. x = 9

47. 2 2 45˚ 45˚ 5 9 5 9 45˚ 45˚ 5 9 Do you notice a pattern?

48. 45˚ x x In a 45˚- 45˚- 90˚ Triangle, hypotenuse = leg 45˚ x 2 2 Theorem 9.8 45˚- 45˚- 90˚ Triangle Theorem

49. y = 7 2 45˚ y x 45˚ 7 3. Find x and y. x = 7

50. 45˚ 3 x 45˚ y 2 4. Find x and y. x = 3 y = 3