Question

given two similar triangles, \\( \triangle abc \sim \triangle def \\), where, \\( b = \frac{4}{5} \\), \\( c = \frac{5}{2} \\), \\( f = \frac{3}{2} \\), find e. if necessary, provide your answer in a completely simplified fraction, like this: 1/2.

Explanation:

Step1: Use the property of similar triangles

For similar triangles \(\triangle ABC\sim\triangle DEF\), the ratios of corresponding sides are equal. So, \(\frac{b}{e}=\frac{c}{f}\).

Step2: Substitute the given values

Substitute \(b = \frac{4}{5}\), \(c=\frac{5}{2}\), and \(f=\frac{3}{2}\) into \(\frac{b}{e}=\frac{c}{f}\). We get \(\frac{\frac{4}{5}}{e}=\frac{\frac{5}{2}}{\frac{3}{2}}\).

Step3: Simplify the right - hand side

Simplify \(\frac{\frac{5}{2}}{\frac{3}{2}}\), which is \(\frac{5}{2}\times\frac{2}{3}=\frac{5}{3}\). So the equation becomes \(\frac{\frac{4}{5}}{e}=\frac{5}{3}\).

Step4: Cross - multiply

Cross - multiply to get \(5e=\frac{4}{5}\times3\).

Step5: Solve for \(e\)

\(5e=\frac{12}{5}\), then \(e=\frac{12}{5}\div5=\frac{12}{5}\times\frac{1}{5}=\frac{12}{25}\).

Answer:

\(\frac{12}{25}\)