Chapter 9 Summary

1. Chapter 9 Summary

2. Similar Right Triangles • If the altitude is drawn to the hypotenuse of a right triangle, then the 3 triangles are all similar.

3. C B A D

4. Find QS.

5. Solve for x

6. C B A D

7. Solve for x

8. Find XZ

9. Pythagorean Theorem • In a right triangle,

10. Acute, Right, Obtuse Triangles • Acute • Right • Obtuse

11. Pythagorean Triples • Any 3 whole numbers that satisfy the pythagorean theorem. • Example: 3, 4, 5 • Nonexample: anything with a decimal or square root!

12. In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. 45°-45°-90° Triangle Theorem 45° x√2 45° Hypotenuse = √2 ∙ leg

13. In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. 30°-60°-90° Triangle Theorem 60° 30° x√3 Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

14. Find the value of x By the Triangle Sum Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x

15. Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2 Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x 45°-45°-90° Triangle Theorem Substitute values Simplify

16. Find the value of x. Because the triangle is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°-90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg. Ex. 2: Finding a leg in a 45°-45°-90° Triangle 5 x x

17. Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem Ex. 2: Finding a leg in a 45°-45°-90° Triangle 5 x x Substitute values 5 √2x = Divide each side by √2 √2 √2 5 = x Simplify √2 Multiply numerator and denominator by √2 √2 5 = x √2 √2 5√2 Simplify = x 2

18. Find the values of s and t. Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. Ex. 3: Finding side lengths in a 30°-60°-90° Triangle 60° 30°

19. Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem Ex. 3: Side lengths in a 30°-60°-90° Triangle 60° 30° Substitute values 5 √3s = Divide each side by √3 √3 √3 5 = s Simplify √3 Multiply numerator and denominator by √3 √3 5 = s √3 √3 5√3 Simplify = s 3

20. Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem The length t of the hypotenuse is twice the length s of the shorter leg. 60° 30° 5√3 t 2 ∙ Substitute values = 3 10√3 Simplify t = 3

21. Parts of the TriangleSohCahToa

22. hypotenuse hypotenuse opposite opposite adjacent adjacent

23. Find the values of the three trigonometric functions of . 5 4 3

24. Finding a missing side

25. Finding a missing angle • We can find an unknown angle in a right triangle, as long as we know the lengths of two of its sides. • Use Trig Inverse • sin-1 • cos-1 • tan-1

26. What is sin-1? But what is the meaning of sin-1… ? Well, the Sine function "sin" takes an angle and gives us the ratio “opposite/hypotenuse” But in this case we know the ratio “opposite/hypotenuse” but want to know the angle. So we want to go backwards. That is why we use sin-1, which means “inverse sine”.

27. Find the missing angle