Question

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1. as shown in the diagram below, $\triangle abc \sim \triangle def$, $ab = 7x$, $bc = 4$, $de = 7$, and $ef = x$. what is the length of $\overline{ab}$?

Explanation:

Step1: Use Similar Triangles Proportion

Since \(\triangle ABC \sim \triangle DEF\), the corresponding sides are proportional. So, \(\frac{AB}{DE}=\frac{BC}{EF}\). Substitute \(AB = 7x\), \(DE = 7\), \(BC = 4\), \(EF = x\) into the proportion: \(\frac{7x}{7}=\frac{4}{x}\).

Step2: Simplify and Solve for \(x\)

Simplify \(\frac{7x}{7}\) to \(x\), so the equation becomes \(x=\frac{4}{x}\). Multiply both sides by \(x\) (assuming \(x
eq0\)): \(x^{2}=4\). Take the square root of both sides: \(x = 2\) (we take the positive root as length is positive).

Step3: Calculate Length of \(AB\)

Now, \(AB = 7x\). Substitute \(x = 2\) into this: \(AB=7\times2 = 14\).

Answer:

14