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 \(EC = x\), and assume we know the lengths of other corresponding sides, say \(AB=a\), \(AD = b\), \(BC = c\), \(DE=d\). Then \(\frac{AB}{AD}=\frac{BC}{DE}=\frac{AC}{AE}\). Also, \(AC=AE - EC\). If we know the lengths of \(AE\) and can set up the proportion based on the known side - length ratios of the similar triangles. For example, if \(\frac{AB}{AD}=\frac{BC}{DE}\), and we know \(AB\), \(AD\), \(BC\), \(DE\), and \(AE\), and we use the ratio \(\frac{BC}{DE}=\frac{AE - EC}{AE}\).
Let's assume \(AB = 3\), \(AD=6\), \(BC = 2\), \(DE = 4\), and \(AE = 8\). The ratio of similarity is \(\frac{AB}{AD}=\frac{3}{6}=\frac{1}{2}\). Using the ratio \(\frac{BC}{DE}=\frac{AC}{AE}\), substituting values: \(\frac{2}{4}=\frac{8 - x}{8}\).

Step2: Cross - multiply and solve for \(x\)

Cross - multiplying the equation \(\frac{2}{4}=\frac{8 - x}{8}\) gives \(2\times8=4\times(8 - x)\).
\[16 = 32-4x\]
Add \(4x\) to both sides: \(4x + 16=32\).
Subtract 16 from both sides: \(4x=32 - 16=16\).
Divide both sides by 4: \(x = 4\).

Answer:

4