Below, PB is tangent to both circles. Angles BO’P and COP are congruent, and PO = 20, OO’ = 40, and OC = 12. What is the length of O'B?

Question

anonymous · Answer

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mathstudent55 · Answer

Look at triangles PBO' and PCO.
Angles BO’P and COP are congruent.
Also, angle P is congruent to angle P.
What can you conclude about the triangles?

anonymous · Answer

that their all congruent?

mathstudent55 · Answer

No, one triangle is larger than the other one, so they are not congruent.
When two angles of one triangle and congruent to two angles of a second triangles, the triangles are similar. This is called proving two triangles similar by AA Similarity.

mathstudent55 · Answer

Once you prove the triangles are similar, like we did, then the lengths of corresponding sides are proportional.

mathstudent55 · Answer

Here are the two similar triangles. I separated them to make it easier to understand.
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mathstudent55 · Answer

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