Question 1159001: Recall that the lengths of the sides of triangle ABC are often abbreviated by writing a = BC, b = CA, and c = AB. Sketch triangle ABC where angle BCA is right and mark F as the foot of the perpendicular drawn from C to the hypotenuse AB. In terms of a, b, and c, express the lengths of FA, FB, and FC. The equation c = FA+FB can be used to check your work.
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website! This is our first triangle: triangle ABC 
When I draw the perpendicular from C to hypotenuse AB, we will have point F, and the first triangle will be split into 2 triangles.
I will labeled them as triangle #2 and triangle #3.
Triangle ABC, triangle #2, and triangle #3 are similar triangles. They have the same shape, the same angle measures, and the same ratios of corresponding sides.
For the ratio of side opposite  to hypotenuse we have:
 --> 
For the ratio of side opposite  to hypotenuse we have:
 --> 
Of course, we know that  ,
We can verify that the expressions we found for  and  are correct, by substituting, and finding that it agrees with what we know:
From  -->  -->  -->  .
However, along the way, we find that we proved the Pythagorean theorem from our knowledge about similar triangles.
We can also prove that  <-->  .
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