Question

△abc and △def are similar.
(a) complete the three pairs of proportions below.
ratio of lengths of sides of △abc
\\( \frac { a b } { a c } = \square \\)
\\( \frac { a b } { b c } = \frac { 2 } { 5 } \\)
\\( \frac { a c } { b c } = \square \\)
ratio of the lengths of sides of △def that correspond to the sides of △abc in the ratio above
\\( \frac {? } { d f } = \frac { 1 } { 2 } \\)
\\( \frac { d e } {? } = \square \\)
\\( \frac {? } { e f } = \frac { 4 } { 5 } \\)
(b) choose the correct statement about the answers to part (a).
○in each pair of proportions, the lengths of the sides of △abc are in the same ratio as the lengths of the sides of △def. this is because we compared the sides of △def that correspond to the sides of △abc and the triangles are similar.
○in each pair of proportions, the lengths of the sides of △abc are in the same ratio as the lengths of the sides of △def. this is coincidence. we would usually not expect this from similar triangles that are not the same size.
○in each pair of proportions, the lengths of the sides of △abc are not in the same ratio as the lengths of the sides of △def. this is because the triangles are not right triangles.
○in each pair of proportions, the lengths of the sides of △abc are not in the same ratio as the lengths of the sides of △def. this is always true for similar triangles.

Explanation:

Step1: Calculate \(\frac{AB}{AC}\)

Given \(AB = 6\) and \(AC=12\), then \(\frac{AB}{AC}=\frac{6}{12}=\frac{1}{2}\).
Since \(\triangle ABC\sim\triangle DEF\), the corresponding side to \(AB\) is \(DE\) (because of similarity, \(AB\) and \(DE\) are corresponding sides). So \(\frac{DE}{DF}=\frac{8}{16}=\frac{1}{2}\).

Step2: Calculate \(\frac{AC}{BC}\)

Given \(AC = 12\) and \(BC = 15\), then \(\frac{AC}{BC}=\frac{12}{15}=\frac{4}{5}\).
Since \(\triangle ABC\sim\triangle DEF\), the corresponding side to \(AC\) is \(DF\) (because of similarity, \(AC\) and \(DF\) are corresponding sides).

Step3: Analyze the proportion \(\frac{AB}{BC}=\frac{2}{5}\)

Given \(AB = 6\) and \(BC = 15\), \(\frac{AB}{BC}=\frac{6}{15}=\frac{2}{5}\).
Since \(\triangle ABC\sim\triangle DEF\), the corresponding side to \(AB\) is \(DE\) and the corresponding side to \(BC\) is \(EF\). So \(\frac{DE}{EF}=\frac{8}{20}=\frac{2}{5}\).

Step4: Analyze the statement in part (b)

For similar triangles \(\triangle ABC\) and \(\triangle DEF\), if \(AB\) corresponds to \(DE\), \(BC\) corresponds to \(EF\) and \(AC\) corresponds to \(DF\), by the property of similar triangles (the ratios of corresponding sides of similar triangles are equal), in each pair of proportions, the lengths of the sides of \(\triangle ABC\) are in the same ratio as the lengths of the sides of \(\triangle DEF\). This is because we compared the sides of \(\triangle DEF\) that correspond to the sides of \(\triangle ABC\) and the triangles are similar.

Answer:

(a)

• \(\frac{AB}{AC}=\frac{1}{2}\), \(\frac{DE}{DF}=\frac{1}{2}\)
• \(\frac{DE}{EF}=\frac{2}{5}\)
• \(\frac{AC}{BC}=\frac{4}{5}\)

(b) In each pair of proportions, the lengths of the sides of \(\triangle ABC\) are in the same ratio as the lengths of the sides of \(\triangle DEF\). This is because we compared the sides of \(\triangle DEF\) that correspond to the sides of \(\triangle ABC\) and the triangles are similar.