Question

δabc and δade are similar. find the length of side \overline{ca}.

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle ABC\) and \(\triangle ADE\) are similar, the ratios of their corresponding sides are equal. That is \(\frac{CA}{EA}=\frac{BA}{DA}\).

We know \(EA = 12\), \(BA=5\), and \(DA=5 + BD\). From the side \(ED = 10\), and because of similarity, we can also use the ratio \(\frac{CA}{EA}=\frac{BC}{ED}\) (another pair of corresponding sides).

Step2: Set up the proportion

Let \(CA=x\). Using \(\frac{CA}{EA}=\frac{BC}{ED}\), substituting the values: \(\frac{x}{12}=\frac{5}{10}\)

Step3: Solve the proportion for \(x\)

Cross - multiply: \(10x=12\times5\)

\(10x = 60\)

Divide both sides by \(10\): \(x=\frac{60}{10}=6\)

Answer:

\(6\)