http://www.ck12.org Chapter 8. Right Triangle Trigonometry
8.3 Using Similar Right Triangles
Learning Objectives
- Identify similar triangles inscribed in a larger triangle.
- Evaluate the geometric mean.
- Find the length of an altitude or leg using the geometric mean.
Review Queue
- If two triangles are right triangles, does that mean they are similar? Explain.
- If two triangles are isosceles right triangles, does that mean they are similar? Explain.
- Solve the ratio:^3 x= 27 x.
- If the legs of an isosceles right triangle are 4, find the length of the hypotenuse. Draw a picture and simplify
the radical.
Know What?In California, the average home price increased 21.3% in 2004 and another 16.0% in 2005. What is
the average rate of increase for these two years?
Inscribed Similar Triangles
You may recall that if two objects are similar, corresponding angles are congruent and their sides are proportional in
length. Let’s look at a right triangle, with an altitude drawn from the right angle.
There are three right triangles in this picture, 4 ADB, 4 CDA, and 4 CAB. Both of the two smaller triangles are
similar to the larger triangle because they each share an angle with 4 ADB. That means all three triangles are similar
to each other.
Theorem 8-5:If an altitude is drawn from the right angle of any right triangle, then the two triangles formed are
similar to the original triangle and all three triangles are similar to each other.
The proof of Theorem 8-5 is in the review questions.
Example 1:Write the similarity statement for the triangles below.