Given \( \triangle A B C \) with vertices at \( -A(1,2), B(3,7) \), and \( C(10,-2) \), and it's dilation, \( \triangle A^{\prime} B^{\prime} C \), with vertices at \( -A^{\prime}(6,12), B^{\prime}(18,42) \), and \( C^{\prime}(60,-12) \). What is the scale factor? Select one: a. \( \frac{1}{3} \) b. \( \frac{1}{2} \) c. 3 d. 6

To find the scale factor of the dilation from triangle \( ABC \) to triangle \( A'B'C \), we can calculate it using the distance between corresponding points. Let's compare the coordinates of points \( A \) and \( A' \): - \( A(-1, 2) \) and \( A'(6, 12) \). To find the scale factor (k), we can use the formula: \[ k = \frac{\text{distance from the origin to } A'}{\text{distance from the origin to } A} \] Calculating these distances using the formula for distance from the origin \(\sqrt{x^2 + y^2}\): For \( A \): \[ \text{Distance to } A = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \] For \( A' \): \[ \text{Distance to } A' = \sqrt{(6)^2 + (12)^2} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5} \] Now, the scale factor is: \[ k = \frac{6\sqrt{5}}{\sqrt{5}} = 6 \] Therefore, the scale factor is 6. The correct answer is: d. 6