Question

triangles abc and def are similar.
a determine the missing values.
b explain your thinking.

Explanation:

Step1: Find the scale factor

Since the triangles are similar, the ratio of corresponding sides is equal. Let's find the scale factor using the sides \(AB = 4\) and \(DE=5\). The scale factor \(k=\frac{DE}{AB}=\frac{5}{4}\)

Step2: Find the length of \(DF\)

We know that \(AC\) corresponds to \(DF\). Let \(AC = x\) and \(DF = 10\). Using the scale factor \(\frac{DF}{AC}=\frac{5}{4}\). Substituting \(DF = 10\), we get \(\frac{10}{x}=\frac{5}{4}\). Cross - multiplying gives \(5x=40\), so \(x = 8\)

Step3: Find the length of \(EF\)

We know that \(BC = 6\) corresponds to \(EF\). Using the scale factor \(\frac{EF}{BC}=\frac{5}{4}\). Substituting \(BC = 6\), we get \(EF=\frac{5}{4}\times6=\frac{30}{4} = 7.5\)

Answer:

\(AC = 8\) and \(EF=7.5\)