Given: Right triangle ABC with altitude Line segment C D Prove: a2 + b2 = c2

Triangle A B C is is shown. Angle A C B is a right angle. Altitude h is drawn from point C to point D on side A B to form a right angle. Line segment A D is labeled e, line segment D B is labeled f, side A B is labeled c, side B C is labeled a, and side A C is labeled b.

Complete the paragraph proof.

You can use the similar triangles formed by the altitude to write ratios for corresponding sides. Using ratios from the large and medium triangles, StartFraction c Over a EndFraction =
. This can be rewritten as
= a2. Using ratios from the large and small triangles,
= StartFraction b Over e EndFraction. 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.