AIR MATH

Math Question

The altitude of a right triangle divides the triangle into two triangles similar to each other and to the original triangle. So, in the diagram below, \( \triangle E F G \sim \triangle F H G \sim \triangle E H F \). Complete the proof that \( E F^{2}+F G^{2}=E G^{2} \). Since \( \triangle E F G \sim \triangle E H F \), then \( \frac{E F}{E G}=\quad \quad \). You can rewrite this equation as -. Similarly, since \( \triangle E F G \sim \triangle F H G \), then \( \frac{E G}{F G}= \) -. You can rewrite this equation as Now, using the - \( E F^{2}+F G^{2}=E F^{2}+E G \cdot H G \). Then, by substitution, \( E F^{2}+F G^{2}= \) -. So, \( E F^{2}+F G^{2}=E G \cdot(E H+H G) \) by the - Because \( E H+H G= \) by the Additive Property of Length, \( E F^{2}+F G^{2}=E G \cdot E G \) using substitution. Therefore, \( E F^{2}+F G^{2}=E G^{2} \).

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