Question

1. the triangles in each pair are similar. find the unknown side lengths.

a) scale factor:
4×scale factor = e
4×______ = e

Explanation:

Step1: Find the scale factor

Since the triangles are similar, the ratio of corresponding sides is the scale factor. The side \(AB = 4\mathrm{cm}\) in \(\triangle ABC\) corresponds to \(DE=8\mathrm{cm}\) in \(\triangle DEF\).
The scale factor \(k=\frac{DE}{AB}=\frac{8}{4} = 2\)

Step2: Find the length of \(e\)

The side \(AC = 5\mathrm{cm}\) in \(\triangle ABC\) corresponds to \(DF = e\) in \(\triangle DEF\).
Using the formula \(e=AC\times k\), substitute \(AC = 5\) and \(k = 2\)
\(e=5\times2=10\mathrm{cm}\)

Step3: Find the length of \(d\)

The side \(BC = 6\mathrm{cm}\) in \(\triangle ABC\) corresponds to \(EF = d\) in \(\triangle DEF\).
Using the formula \(d=BC\times k\), substitute \(BC = 6\) and \(k = 2\)
\(d=6\times2 = 12\mathrm{cm}\)

Answer:

\(d = 12\mathrm{cm}\), \(e=10\mathrm{cm}\)