1. Similar Right Triangles: Geometric Mean

2. Geometric Mean- of two positive numbers a and b is the positive number x such that:– x = b
Geometric Mean is the square root of the product of two positive numbers
Example: The geometric mean of 2 and 8 is:

3. Examples:
Find the geometric mean between:
A and 48
B and 27
C and 54

4. Are these three right triangles similar? Justify your answerC B D
Are these three right triangles similar? Justify your answer
They are similar, by AA~
2. Complete this similarity statement ABC  ____  ____
CBD
ACD

5. 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 length of these two segments C A N B

6. “Heartbeat” C B A N AN CN = NC NB

7. 18 12 y

8. y 5 8

9. 12 16 y

10. Geometric Mean (Leg)Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to the leg. A C B N

11. C A B N

12. C A B N
1ST AB

13. C
2ND A B N
1ST AB BC

14. 3RD C
2ND A B N
1ST AB CB = BC

15. 3RD C
2ND
4TH A B N
1ST AB CB = BC BN

16. A

17. BA A
1ST

18. 2ND BA C AC N A B
1ST

19. 2ND BA CA = AC
3RD
1ST

20. 2ND BA CA = AC AN
3RD
4TH
1ST

21. 27 in.
16 in.
y in.

22. y cm
8 cm.
2 cm

23. 5 ft.
6 ft.
y ft.