1. Geometry 9.4 Special Right Triangles

2. November 27, 2015Geometry 9.4 Special Right Triangles2 Goals I will know the side lengths of special right triangles. I can use special right triangles to solve problems.

3. November 27, 2015Geometry 9.4 Special Right Triangles3 First, radical review: A radical in simplest form has no perfect squares in the radicand.

4. November 27, 2015Geometry 9.4 Special Right Triangles4 Rationalizing Denominators This means no square roots allowed in the denominator of fractions. General rule is to multiply numerator and denominator by the radical and simplify.

5. November 27, 2015Geometry 9.4 Special Right Triangles5 Examples

6. November 27, 2015Geometry 9.4 Special Right Triangles6

7. November 27, 2015Geometry 9.4 Special Right Triangles7 Constructing Special Triangles Construct an equilateral triangle. Each angle measures ______. 60 

8. November 27, 2015Geometry 9.4 Special Right Triangles8 Constructing Special Triangles Draw an altitude. It is perpendicular to the base. It also bisects the vertex angle. 60  30 

9. November 27, 2015Geometry 9.4 Special Right Triangles9 Constructing Special Triangles Clean up the drawing – only keep the triangle on the left. 60  30 

10. November 27, 2015Geometry 9.4 Special Right Triangles10 Constructing Special Triangles This is called a 30 – 60 – 90 triangle. 60  30 

11. November 27, 2015Geometry 9.4 Special Right Triangles11 Constructing Special Triangles Give the original side an arbitrary length. Call it 2a. 60  30  2a

12. November 27, 2015Geometry 9.4 Special Right Triangles12 Constructing Special Triangles What is the length of the base? 60  30  2a ? a

13. November 27, 2015Geometry 9.4 Special Right Triangles13 Constructing Special Triangles Now find the height of the triangle, h. a 2 + h 2 = (2a) 2 60  30  2a a h

14. November 27, 2015Geometry 9.4 Special Right Triangles14 Constructing Special Triangles 60  30  2a a h

15. November 27, 2015Geometry 9.4 Special Right Triangles15 30-60-90 Triangle 60  30  2a2a a

16. November 27, 2015Geometry 9.4 Special Right Triangles16 Example 1 Find x & y. 60  30  2a2a a 60  30  12 x y 6

17. November 27, 2015Geometry 9.4 Special Right Triangles17 Example 2 Find x & y. 60  30  2a2a a 60  30  8 x y 16

18. November 27, 2015Geometry 9.4 Special Right Triangles18 Example 3 Find x & y. 60  30  2a2a a 60  30  40 x y 20

19. November 27, 2015Geometry 9.4 Special Right Triangles19 Example 4 Find x & y. 60  30  2a2a a 60  30  x y 12

20. November 27, 2015Geometry 9.4 Special Right Triangles20 Example 4 Solution 60  30  x y 12

21. November 27, 2015Geometry 9.4 Special Right Triangles21 30-60-90 Triangle 60  30  2a2a a Learn the pattern!

22. November 27, 2015Geometry 9.4 Special Right Triangles22 Try it. Find x and y. 60  30  x 15 y 30

23. November 27, 2015Geometry 9.4 Special Right Triangles23 Another one: Find h and y. 60  30  y h 14 7

24. November 27, 2015Geometry 9.4 Special Right Triangles24 Again: Find x and y. 60  30  y x 2 4

25. November 27, 2015Geometry 9.4 Special Right Triangles25 Drawing Special Triangles Draw a Square. Draw one diagonal. The diagonal bisects the angles.  1 &  2 measure ______. 45  1 2

26. November 27, 2015Geometry 9.4 Special Right Triangles26 Drawing Special Triangles Clean it up – keep only the triangle on the right. 45 

27. November 27, 2015Geometry 9.4 Special Right Triangles27 Constructing Special Triangles In the original square, each side was the same. Label them a. 45  a a

28. November 27, 2015Geometry 9.4 Special Right Triangles28 Constructing Special Triangles Solve for the hypotenuse, c. 45  a a c

29. November 27, 2015Geometry 9.4 Special Right Triangles29 45-45-90 Triangle 45  a a

30. November 27, 2015Geometry 9.4 Special Right Triangles30 Example 5 Solve for x & y. 45  5 x y a a a2a2 5

31. November 27, 2015Geometry 9.4 Special Right Triangles31 Example 6 Solve for x & y. 45  8 x y a a a2a2 8

32. November 27, 2015Geometry 9.4 Special Right Triangles32 Example 7: A tougher one. Solve for x & y. 45  x10 y a 45  a a2a2

33. November 27, 2015Geometry 9.4 Special Right Triangles33 Example 7 Solution 45  x10 y

34. November 27, 2015Geometry 9.4 Special Right Triangles34 45-45-90 Triangle 45  a a Learn the pattern!

35. November 27, 2015Geometry 9.4 Special Right Triangles35 Try it. Find x and y. 45  x y 8

36. November 27, 2015Geometry 9.4 Special Right Triangles36 45-45-90 Triangle 45  a Shortcut pattern:

37. November 27, 2015Geometry 9.4 Special Right Triangles37 Example 8Solve for x & y 40 x y

38. November 27, 2015Geometry 9.4 Special Right Triangles38 Example 9 A designer wants to put a rope light on the diagonal of a square dance floor. If the floor measures 30 ft. on a side, how long does the rope light need to be?

39. November 27, 2015Geometry 9.4 Special Right Triangles39 Example 9 Solution 30 ft. 45 

40. November 27, 2015Geometry 9.4 Special Right Triangles40 Summary 60  30  2a2a a 45  a a 45-45-9030-60-90 In Trigonometry, you will understand why these triangle are so important.

41. November 27, 2015Geometry 9.4 Special Right Triangles41 Quick Quiz – 5 problems 30  x y 12 1. Find x & y. 24

42. November 27, 2015Geometry 9.4 Special Right Triangles42 60  x y 50 2. Find x & y. 25

43. November 27, 2015Geometry 9.4 Special Right Triangles43 75 60  x 3. Find the area of the triangle.

44. November 27, 2015Geometry 9.4 Special Right Triangles44 75 60  25  3 3. Find the area of the triangle.

45. November 27, 2015Geometry 9.4 Special Right Triangles45 4. Find x. 62 x 45 

46. November 27, 2015Geometry 9.4 Special Right Triangles46 5. Find the area. 45  7 7

47. November 27, 2015Geometry 9.4 Special Right Triangles47 One More Time… 60  30  2a2a a 45  a a a2a2