You can use the similar triangles formed by the altitude to write ratios for corresponding sides. Using ratios from the large and medium triangles, = . This can be rewritten as = a2. Using ratios from the large and small triangles, = . This can be rewritten as b2 = ec. By substitution, a2 + b2 =. You can then factor as a2 + b2 = c(f + e). From the large triangle, you know (f + e) =. So, a2 + b2 = c2 by using substitution.
The altitude creates two smaller triangles similar to the original right triangle. The resulting leg-to-hypotenuse ratios give a² = cf and b² = ce, so adding them gives a² + b² = cf + ec. This matches choice A, “cf + ec”; factoring gives c(f + e), and f + e is the full hypotenuse c.
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