Question

in the figure to the right, △abc and △ade are similar. find the length of (overline{ec}). the length of (overline{ec}) is (simplify your answer.)

Explanation:

Step1: Set up proportion from similar - triangles

Since \(\triangle ABC\) and \(\triangle ADE\) are similar, the ratios of their corresponding sides are equal. Let \(EC = x\), then \(AC=1 + x\) and \(AE = 7\), \(AB = 1\), \(AD=6\). The proportion is \(\frac{AB}{AD}=\frac{AC}{AE}\), so \(\frac{1}{6}=\frac{1 + x}{7}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{1}{6}=\frac{1 + x}{7}\) gives \(7\times1=6\times(1 + x)\).

Step3: Expand and solve for \(x\)

Expand the right - hand side: \(7 = 6+6x\). Subtract 6 from both sides: \(7 - 6=6x\), so \(1 = 6x\). Then \(x=\frac{1}{6}\).

Answer:

\(\frac{1}{6}\)