Question

given: \\( \triangle abc \\) is a right triangle and \\( \overline{cd} \perp \overline{ab} \\).
prove: \\( ac^{2}+bc^{2}=ab^{2} \\)

proof:

which reason completes the proof?

Explanation:

Step1: Similar Triangles Proportional Sides

When two triangles are similar, the ratios of their corresponding sides are equal. In step 3, we have \(\triangle ABC\sim\triangle ACD\) and \(\triangle ABC\sim\triangle CBD\). For similar triangles \(\triangle ABC\) and \(\triangle ACD\), the proportion \(\frac{AC}{AB}=\frac{AD}{AC}\) comes from the fact that in similar triangles, the ratios of corresponding sides are equal. Similarly, for similar triangles \(\triangle ABC\) and \(\triangle CBD\), \(\frac{BC}{AB}=\frac{DB}{BC}\) is based on the property of similar - triangle side - ratios.

Answer:

Corresponding sides of similar triangles are proportional.