In the coordinate plane, a dilation centered at the origin maps point $P(2,5)$ to $P\prime (6,15)$. What is the scale factor of the dilation?

• A. $\frac{3}{2}$
• B. $2$
• C. $3$
• D. $\frac{5}{15}$

Answer: (C)

Verified step by step guidance

1. Recall that a dilation centered at the origin multiplies both the x- and y-coordinates of a point by the same scale factor, say \(k\).
2. Set up the equation for the x-coordinate: \(k \times 2 = 6\) because the original point is \((2, 5)\) and the image is \((6, 15)\).
3. Solve for \(k\) from the x-coordinate equation: \(k = \frac{6}{2}\).
4. Verify the scale factor by checking the y-coordinate: \(k \times 5 = 15\).
5. Confirm that the scale factor \(k\) is consistent for both coordinates, which means the dilation scale factor is \(k\).