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 using similar - triangles property

Since \(\triangle ABC\) and \(\triangle ADE\) are similar, the ratios of corresponding sides are equal. Let \(EC = x\), then \(AC=x + 1\) and \(AE=x + 1+7=x + 8\). The ratio of the sides is \(\frac{AC}{AE}=\frac{BC}{DE}\). We know that \(BC = 1\), \(DE = 6\), \(AC=x + 1\), and \(AE=x + 8\). So, \(\frac{x + 1}{x + 8}=\frac{1}{6}\).

Step2: Cross - multiply the proportion

Cross - multiplying the proportion \(\frac{x + 1}{x + 8}=\frac{1}{6}\) gives us \(6(x + 1)=x + 8\).

Step3: Expand the left - hand side

Expanding \(6(x + 1)\) using the distributive property \(a(b + c)=ab+ac\) (where \(a = 6\), \(b=x\), \(c = 1\)), we get \(6x+6=x + 8\).

Step4: Solve for \(x\)

Subtract \(x\) from both sides: \(6x - x+6=x - x + 8\), which simplifies to \(5x+6 = 8\). Then subtract 6 from both sides: \(5x+6 - 6=8 - 6\), so \(5x=2\). Divide both sides by 5: \(x=\frac{2}{5}=0.4\).

Answer:

\(0.4\)