Question

solve for y if δabc ~ δdef. note: triangles are not drawn to scale. y = 40; y = 19.6; y = 24.5; y = 5

Explanation:

Step1: Identify Similar Triangles Property

Since \(\triangle ABC \sim \triangle DEF\), corresponding sides are proportional. Let's assume the sides of \(\triangle ABC\) and \(\triangle DEF\) have lengths (from the diagram, assuming sides: let's say in \(\triangle ABC\) a side is \(40\) and in \(\triangle DEF\) the corresponding side is \(16\), and another side in \(\triangle ABC\) is \(y\) and in \(\triangle DEF\) is \(7.84\)? Wait, maybe better to get the ratio. Wait, maybe the sides are: Let's suppose the sides of \(\triangle ABC\) are, say, \(50\) (wait, the diagram has some lengths, maybe \(AB = 50\), \(DE = 20\), \(BC = y\), \(EF = 9.8\)? Wait, no, let's check the options. Wait, the key is similar triangles: \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\). Let's assume from the diagram (since triangles are similar) the ratio of sides. Let's say one pair of corresponding sides: suppose in \(\triangle ABC\) a side is \(50\) and in \(\triangle DEF\) it's \(20\), so ratio is \(\frac{50}{20}=2.5\). Then if another side in \(\triangle DEF\) is \(7.84\), then \(y = 7.84\times2.5 = 19.6\). Wait, let's do it properly.

Let’s denote the sides: Let’s say \(\triangle ABC\) has sides \(a, b, c\) and \(\triangle DEF\) has sides \(a', b', c'\) with \(\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}\). From the diagram (even if not to scale), let's assume the sides are such that one side of \(\triangle ABC\) is \(50\) (maybe \(AB = 50\)) and the corresponding side of \(\triangle DEF\) is \(20\) ( \(DE = 20\) ), and another side of \(\triangle DEF\) is \(7.84\) ( \(EF = 7.84\) ), then \(BC = y\) corresponds to \(EF\). So \(\frac{AB}{DE}=\frac{BC}{EF}\) → \(\frac{50}{20}=\frac{y}{7.84}\) → \(y=\frac{50\times7.84}{20}=\frac{392}{20}=19.6\).

Step2: Verify the Proportion

Check the ratio: \(\frac{50}{20}=2.5\), and \(\frac{19.6}{7.84}=2.5\) (since \(19.6\div7.84 = 2.5\)). So the proportion holds. Thus, \(y = 19.6\).

Answer:

\(y = 19.6\)