Question

given two similar triangles, ( \triangle abc sim \triangle def ), where, ( d = 5 ), ( e = 2.5 ), ( b = 7.5 ), find ( a ). if necessary, round to the tenths place. question 3 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. That is \(\frac{a}{d}=\frac{b}{e}\).

Step2: Substitute the given values

We know \(d = 5\), \(e=2.5\), \(b = 7.5\). Substituting into \(\frac{a}{d}=\frac{b}{e}\), we get \(\frac{a}{5}=\frac{7.5}{2.5}\).

Step3: Solve for \(a\)

Cross - multiply: \(a\times2.5=7.5\times5\). Then \(a=\frac{7.5\times5}{2.5}\). Since \(7.5\times5 = 37.5\) and \(37.5\div2.5=15\).

Step4: For the second problem

Since \(\triangle ABC\sim\triangle DEF\), \(\frac{b}{e}=\frac{c}{f}\). Given \(b=\frac{4}{5}\), \(c = \frac{5}{2}\), \(f=\frac{3}{2}\). Substitute into \(\frac{b}{e}=\frac{c}{f}\), we have \(\frac{\frac{4}{5}}{e}=\frac{\frac{5}{2}}{\frac{3}{2}}\).

Step5: Simplify the right - hand side

\(\frac{\frac{5}{2}}{\frac{3}{2}}=\frac{5}{3}\). So \(\frac{4}{5e}=\frac{5}{3}\).

Step6: Cross - multiply

Cross - multiply gives \(4\times3 = 5\times5e\). So \(12 = 25e\). Then \(e=\frac{12}{25}\).

Answer:

15
\(\frac{12}{25}\)