Question

in the figure to the right, △abc and △ade are similar. find the length of ec. the length of 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 \(AC=x + 1\). We have the proportion \(\frac{BC}{DE}=\frac{AC}{AD}\). Given \(BC = 1\), \(DE = 6\), \(AD=3\), and \(AC=x + 1\), the proportion is \(\frac{1}{6}=\frac{x + 1}{3}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{1}{6}=\frac{x + 1}{3}\) gives \(3\times1=6\times(x + 1)\).

Step3: Expand and solve for \(x\)

Expand the right - hand side: \(3 = 6x+6\). Subtract 6 from both sides: \(3-6=6x\), so \(- 3=6x\). Then divide both sides by 6: \(x=-\frac{3}{6}=-\frac{1}{2}\). But length cannot be negative. There is a mistake above. The correct proportion should be \(\frac{BC}{DE}=\frac{AB}{AD}\). Let \(EC=x\), then \(AC = 1\), \(AD=3\), \(BC = 1\), \(DE = 6\). The correct proportion is \(\frac{1}{6}=\frac{1}{3}\) (wrong). Let's start over. Since \(\triangle ABC\sim\triangle ADE\), we have \(\frac{BC}{DE}=\frac{AC}{AE}\). Let \(EC=x\), then \(AE=x + 6\), \(AC = 1\), \(BC = 1\), \(DE = 6\). So \(\frac{1}{6}=\frac{1}{x + 6}\). Cross - multiplying gives \(x+6 = 6\), then \(x=0\).
Let's assume the correct proportion is \(\frac{BC}{DE}=\frac{AC}{AD}\). Given \(BC = 1\), \(DE = 6\), \(AD = 3\), and \(AC\) is related to \(EC\). Let \(EC=x\), then \(AC\) and \(AE\) are involved. Since \(\triangle ABC\sim\triangle ADE\), we know that \(\frac{BC}{DE}=\frac{AC}{AD}\). Let \(EC=x\), \(AC = 1\), \(AD = 3\), \(BC = 1\), \(DE = 6\). The correct proportion is \(\frac{BC}{DE}=\frac{AC}{AD}\). We know that \(\frac{1}{6}=\frac{AC}{3}\), so \(AC=\frac{1}{2}\). If \(AC=\frac{1}{2}\) and \(AE = 6\), then \(EC=AE - AC\). Since \(AC=\frac{1}{2}\) and \(AE = 6\), \(EC=6-\frac{1}{2}=\frac{12 - 1}{2}=\frac{11}{2}\).

Answer:

\(\frac{11}{2}\)