Question

pentagon k is a scaled copy of pentagon j.
pentagon j has sides labeled 20, 40, 24.
pentagon k has sides labeled 15, 30, 18.
what scale factor takes pentagon j to pentagon k?

Explanation:

Step1: Recall the formula for scale factor

The scale factor \(k\) from pentagon \(J\) to pentagon \(K\) is given by \(k=\frac{\text{Length in }K}{\text{Length in }J}\).

Step2: Choose a pair of corresponding sides

Let's use the sides of length \(20\) (in \(J\)) and \(15\) (in \(K\)). Then \(k = \frac{15}{20}\).

Step3: Simplify the fraction

\(\frac{15}{20}=\frac{15\div5}{20\div5}=\frac{3}{4}\).

We can check with another pair of sides. For example, sides of length \(40\) (in \(J\)) and \(30\) (in \(K\)): \(\frac{30}{40}=\frac{30\div10}{40\div10}=\frac{3}{4}\). And for sides of length \(24\) (in \(J\)) and \(18\) (in \(K\)): \(\frac{18}{24}=\frac{18\div6}{24\div6}=\frac{3}{4}\).

Answer:

\(\frac{3}{4}\)