1. SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER

2. How much do you know

3. DRILL
SIMPLIFY THE FOLLOWING EXPRESSION. + 2. 5. 3.

4. DRILL Find the geometric mean between the two given numbers.
and 8
and 4

5. DRILL
Find the geometric mean between the two given numbers.
6 and 8 h= =
h= 4

6. DRILL
Find the geometric mean between the two given numbers.
and 4 h= =
h= 6

7. REVIEW ABOUT RIGHT TRIANGLES
LEGS A
&
The perpendicular side
HYPOTENUSE B C
The side opposite the right angle

8. SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER

9. CONSIDER THIS…
State the means and the extremes in the following statement.
3:7 = 6:14
The means are 7 and 6 and the extremes are 3 and 14.

10. CONSIDER THIS…
State the means and the extremes in the following statement.
5:3 = 6:10
The means are 3 and 6 and the extremes are 5 and 10.

11. a:h = h:b State the means and the extremes in the following statement.
CONSIDER THIS…
State the means and the extremes in the following statement.
a:h = h:b
The means are h and the extremes are a and b.

12. applying the law of proportion.
CONSIDER THIS…
Find h.
a:h = h:b
applying the law of proportion.
h² = ab h=
h is the geometric mean between a & b.

13. THEOREM: SIMILARITIES IN A RIGHT TRIANGLE
States that “In a right triangle, the altitude to the hypotenuse separates the triangle into two triangles each similar to the given triangle and to each other.

14. ∆MOR ~ ∆MSO, ∆MOR ~ ∆OSR by AA Similarity postulate) ILLUSTRATION
“In a right triangle (∆MOR), the altitude to the hypotenuse(OS) separates the triangle into two triangles(∆MOS & ∆SOR )each similar to the given triangle (∆MOR) and to each other.
∆MSO~ ∆OSR by transitivity

15. TRY THIS OUT! ∆ACD ~ ∆ABC ∆ACD ~ ∆CBD ∆ABC ~ ∆CBD D B C A
NAME ALL SIMILAR TRIANGLES
∆ACD ~ ∆ABC
∆ACD ~ ∆CBD
∆ABC ~ ∆CBD

16. In a right triangle, the altitude to the hypotenuse is the geometric
COROLLARY 1.
In a right triangle, the altitude to the hypotenuse is the geometric
mean of the segments into which it divides the hypotenuse

17. ILLUSTRATION In the figure, DB D C
CB is the geometric mean between AB & BD.
In the figure,

18. COROLLARY 2.
In a right triangle, either leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to it.

19. ILLUSTRATION In the figure, DB D C
CB is the geometric mean between AB & BD.
In the figure,