This document discusses properties of right triangles and similarity. It contains three main points: 1) The altitude to the hypotenuse of a right triangle separates it into two smaller right triangles that are all similar to each other. 2) The altitude is the geometric mean of the hypotenuse segments and each leg is the geometric mean of its adjacent hypotenuse segment and the hypotenuse. 3) Examples demonstrate using proportions to solve for missing side lengths in similar right triangles. Exercises provide additional practice applying the concepts.

1. 48 TRIANGLES

2. Formula
Let n = number of rows
If n is even:
Total no. of Δ푠 =
푛(푛+2)(2푛+1)
8
If n is odd:
Total no. of Δ푠 =
푛+1 (2푛2+3푛 −1)
8

3. n = 5
Total no. of Δ푠 =
푛+1 (2푛2+3푛 −1)
8
Total no. of Δs =
5+1 [2(5)2+3 5 −1]
8
=
6 (50+14)
8
6(64)
=
8
=
384
8
= 48

4. SIMILARITIES IN
RIGHT TRIANGLES

5. Theorem 1
The altitude to the hypotenuse
of a right triangle separates
the right triangle into two
triangles which are similar to
each other and to the original
triangle.

6. Example
In a right Δ퐴퐵퐶, 퐵퐸 is an altitude.

7. Three similar triangles:
Δ푨푩푪 ~ Δ푨푬푩 ~ Δ푩푬푪
Congruent angles:
BEC ≅ AEB ≅ ABC
A ≅ EBC
ABE ≅ ECB

8. Theorem 2
In any right triangle,
a.The altitude to the hypotenuse is the
geometric mean between the
segments into which it separates the
hypotenuse.
b.Each leg is a geometric mean of the
hypotenuse and the segment of the
hypotenuse adjacent to the leg.

9. B
A E C
Three pairs of similar triangles:
 ΔAEB ~ ΔBEC
 ΔBEC ~ ΔABC
 ΔAEB ~ ΔABC

10. Proportions:
a.ΔAEB ~ ΔBEC,
푨푬
푩푬
=
푩푬
푪푬
b.ΔBEC ~ ΔABC,
푨푪
푪푩
=
푪푩
푪푬
c.ΔAEB ~ ΔABC,
푨푪
푨푩
=
푨푩
푨푬

11. 푆푖푑푒
퐴푙푡푖푡푢푑푒
=
퐴푙푡푖푡푢푑푒
푆푖푑푒
퐻푦푝표푡푒푛푢푠푒
퐿푒푔
=
퐿푒푔
푆푖푑푒

12. Example 1
Given: s1 = 3, s2 = 9
Unknown: altitude h

13. Example 2
Solve for side x of bigΔ.

14. Example 3
Solve for side x of smallΔ.

15. Exercises
Right ΔRAE, with 퐴푃 an altitude.
R
P
A E
1.If RP = 3 and PE = 8, find AP and AR.
2.If AP = 8 and PE = 12, find RE and AE.

16. Generalization
 Two triangles are similar if their corresponding angles are
congruent.
 The altitude to the hypotenuse of a right triangle separates
the right triangle into two triangles which are similar to each
other and to the original triangle.
 In any right triangle, the altitude to the hypotenuse is the
geometric mean between the segments into which it
separates the hypotenuse, and each leg is a geometric mean
of the hypotenuse and the segment of the hypotenuse
adjacent to the leg.

17. Assignment
Finding the Height of a
Roof:
A roof has a cross section
that is a right angle. The
diagram shows the
approximate dimensions
of this cross section.
A. Identify the similar
triangles.
B. Find the height h of
the roof.

18. THE END