Problem 4 Quadrilateral \( A \) has side lengths \( 2,3,5 \), and 6 . Quadrilateral \( B \) has side lengths \( 4,5,8 \), and 10. Could one of the quadrilaterals be a scaled copy of the other? Explain.

To determine if one quadrilateral could be a scaled copy of the other, we need to check if the ratios of the corresponding side lengths are the same. Let's denote the side lengths of quadrilateral \( A \) as \( a_1 = 2 \), \( a_2 = 3 \), \( a_3 = 5 \), and \( a_4 = 6 \). The side lengths of quadrilateral \( B \) are \( b_1 = 4 \), \( b_2 = 5 \), \( b_3 = 8 \), and \( b_4 = 10 \). First, we will calculate the ratios of the corresponding sides of quadrilaterals \( A \) and \( B \): 1. For \( a_1 \) and \( b_1 \): \[ \frac{b_1}{a_1} = \frac{4}{2} = 2 \] 2. For \( a_2 \) and \( b_2 \): \[ \frac{b_2}{a_2} = \frac{5}{3} \approx 1.67 \] 3. For \( a_3 \) and \( b_3 \): \[ \frac{b_3}{a_3} = \frac{8}{5} = 1.6 \] 4. For \( a_4 \) and \( b_4 \): \[ \frac{b_4}{a_4} = \frac{10}{6} \approx 1.67 \] Now, we can summarize the ratios: - \( \frac{b_1}{a_1} = 2 \) - \( \frac{b_2}{a_2} \approx 1.67 \) - \( \frac{b_3}{a_3} = 1.6 \) - \( \frac{b_4}{a_4} \approx 1.67 \) Since the ratios of the corresponding sides are not all equal, quadrilateral \( A \) cannot be a scaled copy of quadrilateral \( B \). Thus, the conclusion is that **one of the quadrilaterals cannot be a scaled copy of the other** because the ratios of their corresponding side lengths are not consistent.