1. 9.1 Similar Right Triangles
Unit IIB Day 1

2. Do now: Fill in the blank.
In similar triangles, corresponding side lengths are _________________
Corresponding angles are _______________
What is the definition of the altitude of a triangle?

3. Investigating Similar Right Triangles
Cut an index card along one of its diagonals.
On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two more right triangles.
You should now have three right triangles. Compare the angles of your three triangles by overlapping corresponding parts.
What might you conclude about the three triangles?
If the 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

4. Theorem 9.1
If the 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.
∆CBD ~ ∆ABC ~ ∆ACD
We will call the sides of right triangles the long leg, short leg, and hypotenuse.

5. Informal Proof of thm. 9.1:
GIVEN: ∆ABC is a right triangle; altitude CD is drawn to hypotenuse AB.
PROVE: ∆CBD ~ ∆ABC ~ ∆ACD
First prove that ∆CBD ~ ∆ABC.
Each triangle has a right angle, and each includes B (aka 2)
AA Similarity Postulate.
Similarly, ∆ACD ~ ∆ABC.
To prove ∆CBD ~ ∆ACD…
3  2 because both are complementary to 1. (congruent complements thm)
Then you can use the AA Similarity Postulate.

6. Ex. 1: Finding the Height of a Roof
A roof has a cross section that is a right angle.
Identify the similar triangles.
Find the height h of the roof.

7. Ex. 1 Redraw the triangles… a) ∆XYW ~ ∆YZW ~ ∆XZY.
b) Use the fact that ∆XYW ~ ∆XZY to write a proportion.
YW/ZY = XY/XZ
h/5.5 = 3.1/6.3
h ≈ 2.7 m

8. Ex. 1A
The diagram shows the approximate dimensions of a right triangle.
Identify the similar triangles in the diagram.
Find the length of the altitude, h.
ΔRSU ~ ΔSTU ~ ΔRTS
6.0 inches

9. Ex. 2 Fill in the question mark. Then solve for x. a) b)
The proportion is long leg/short leg, so 12 completes the proportion. x = 18
The proportion is short leg/hypotenuse, so 20 completes the proportion. x = 6√5