What if you drew a line from the right angle of a right triangle perpendicular to the side that is opposite that angle? How could you determine the length of that line? After completing this Concept, you'll be able to solve problems like this one.
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CK12 Foundation: Inscribed Similar Triangles
Guidance
Remember that if two objects are similar, their corresponding angles are congruent and their sides are proportional in length. The altitude of a right triangle creates similar triangles.
Inscribed Similar Triangles Theorem: 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.
In
So,
This means that all of the corresponding sides are proportional. You can use this fact to find missing lengths in right triangles.
Example A
Find the value of
Separate the triangles to find the corresponding sides.
Set up a proportion.
Example B
Find the value of
Set up a proportion.
Example C
Find the values of
Separate the triangles. Write a proportion for
Set up a proportion for
CK12 Foundation: Inscribed Similar Triangles
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Guided Practice
1. Find the value of
2. Now find the value of
3. Write the similarity statement for the right triangles in the diagram.
Answers:
1. Set up a proportion.
2. Use the Pythagorean Theorem.
3.
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Fill in the blanks.

△BAD∼△−−−−−∼△−−−−− 
BC?=?CD 
BCAB=AB? 
?AD=ADBD
Write the similarity statement for the right triangles in each diagram.
Use the diagram to answer questions 710.
 Write the similarity statement for the three triangles in the diagram.
 If
JM=12 andML=9 , findKM .  Find
JK .  Find
KL .
Find the length of the missing variable(s). Simplify all radicals.
 Fill in the blanks of the proof for the Inscribed Similar Triangles Theorem.
Given:
Prove:
Statement  Reason 

1.  1. Given 
2. 
2. 
3. 
3. 
4.  4. Reflexive PoC 
5.  5. AA Similarity Postulate 
6. 
6. 
7. 
7. 
8. 
8. 