On a coordinate plane, the point $P(2,3)$ on a figure is dilated about the origin to $P\prime (6,9)$. What is the scale factor of the dilation?

• A. $3$
• B. $\frac{1}{3}$
• C. $2$
• D. $6$

Answer: (A)

Verified step by step guidance

1. Identify the original point and its image after dilation. The original point is \(P(2, 3)\) and the image point is \(P\prime(6, 9)\).
2. Recall that dilation about the origin scales both the x- and y-coordinates of a point by the same scale factor \(k\). This means \(P\prime = (k \cdot x, k \cdot y)\) where \((x, y)\) are the coordinates of the original point.
3. Set up equations for the x- and y-coordinates using the scale factor \(k\): \$6 = k \cdot 2\( and \)9 = k \cdot 3$.
4. Solve each equation for \(k\): from \$6 = 2k\(, we get \)k = \frac{6}{2}\(; from \)9 = 3k\(, we get \)k = \frac{9}{3}$.
5. Verify that both values of \(k\) are equal to confirm the scale factor of the dilation.