Proof 5: Using similar shapes and area ratios Take a right-angled triangle and draw an altitude (line from the vertex C which is perpendicular to the base AB). C.
b
a
A c
D
B
Triangles CBD, ADC and ABC are similar and their length ratio is a : b : c Therefore their areas are in the ratio a 2 : b 2 : c 2 So we can call the areas ka 2 , kb 2 and kc 2 for some constant, k. But clearly area CBD + area ADC = area ABC, so ka 2 + kb 2 = kc 2 and a 2 + b 2 = c 2 . RDS
