9.3 Altitude-on-Hypotenuse Theorems

1. 9.3 Altitude-on-Hypotenuse Theorems Objectives: 1. To find the geometric mean of two numbers. 2. To find missing lengths of similar right triangles that result when an altitude is drawn to the hypotenuse

2. Altitudes Recall that an altitude is a segment drawn from a vertex such that it is perpendicular to the opposite side of the triangle. Every triangle has three altitudes.

3. Altitudes In a right triangle, two of these altitudes are the two legs of the triangle. The other one is drawn perpendicular to the hypotenuse. Altitudes:

4. Altitudes Notice that the third altitude creates two smaller right triangles. Is there something special about the three triangles? Altitudes:

5. Index Card Activity Step 1: On your index card, draw a diagonal. Cut along this diagonal to separate your right triangles.

6. Index Card Activity Step 2: On one of the right triangles, fold the paper to create the altitude from the right angle to the hypotenuse. Cut along this fold line to separate your right triangles.

7. Index Card Activity Step 2: On one of the right triangles, fold the paper to create the altitude from the right angle to the hypotenuse. Cut along this fold line to separate your right triangles.

8. Index Card Activity Step 2: On one of the right triangles, fold the paper to create the altitude from the right angle to the hypotenuse. Cut along this fold line to separate your right triangles.

9. Index Card Activity Step 3: Notice that the small and medium triangle can be stacked on the large triangle so that the side they share is the third altitude of the large triangle.

10. Index Card Activity Step 3: Label the angles of each triangle to match the diagram. Be sure to label all three triangles. In fact, use 4 colored markers to color edges and altitude, and label the back of each of them too. B A C

11. Index Card Activity Step 4: Arrange all three triangles so that they are nested. What does this demonstrate? Why must this be true? C

12. Index Card Activity Step 5: Write a similarity statement involving the large, medium, and small triangles. C

13. Right Triangle Similarity Theorem If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

14. Example 1 Identify the similar triangles in the diagram.

15. Example 2 Find the value of x. Did you get x = 12 ? 5

16. Geometric Mean The geometric mean of two positive numbers a and b is the positive number x that satisfies This is just the square root of their product!

17. Example 3 Find the geometric mean of 12 and 27.

18. Example 4 Find the value of x. Did you get x = 18?

19. Example 5 The altitude to the hypotenuse divides the hypotenuse into two segments. What is the relationship between the altitude and these two segments? altitude Segment 1 Segment 2 hypotenuse

20. Geometric Mean Theorem I Geometric Mean (Altitude) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

21. Geometric Mean Theorem I Heartbeat Geometric Mean (Altitude) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. x x b a

22. Example 6 Find the value of w. (w + 9)(w + 9) = (8)(18) w2 + 18w + 81 = 144 (short leg) 8 w + 9 = (long leg) w + 9 18 w2 + 18w - 63 = 0 (w + 21)(w – 3) = 0 w = {-21, 3} 8 : (3 + 9) = (3 + 9) : 18 8 : 12 = 12 : 18 2 : 3 = 2 : 3 √

23. Example 7 Find the value of x. Boomerang! (short leg) 3 x = (hypotenuse) x 12 x2 = 36 Did you get x = 6?

24. Geometric Mean Theorem II Geometric Mean (Leg) Theorem When an altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

25. Geometric Mean Theorem II Boomerang Geometric Mean (Leg) Theorem When an altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. a a x c

26. Geometric Mean Theorem II Boomerang Geometric Mean (Leg) Theorem When an altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. b b y c

27. Example 8 Find the value of b. Did you get b = 25 ? 6

28. Example 9 Find the value of the variable. 6 4 Boomerang! HEARTBEAT

29. 9.3 Assignment P 379: (1 – 5; 8 – 13; 16, 17)