Question

ab has endpoints a(2, 9) and b(5, 8). a dilation, centered at the origin, is applied to ab. the image has endpoints a($\frac{4}{3}$, 6) and b($\frac{10}{3}$, $\frac{16}{3}$). what is the scale factor of this dilation?
○ $\frac{2}{3}$
○ $\frac{3}{2}$
○ 2
○ 3

Explanation:

Step1: Recall dilation formula

For a dilation centered at the origin, if a point $(x,y)$ is dilated to $(x',y')$, the scale - factor $k$ can be found by $x' = kx$ and $y' = ky$. We can use either the $x$ - coordinates or the $y$ - coordinates of the pre - image and image points. Let's use the $x$ - coordinates of points $A$ and $A'$.
For point $A(2,9)$ and $A'(\frac{4}{3},6)$, we have $x = 2$ and $x'=\frac{4}{3}$.

Step2: Calculate the scale factor

Using the formula $x' = kx$, we substitute $x = 2$ and $x'=\frac{4}{3}$ into it: $\frac{4}{3}=k\times2$. Solving for $k$, we get $k=\frac{\frac{4}{3}}{2}=\frac{4}{3}\times\frac{1}{2}=\frac{2}{3}$.
We can double - check with the $y$ - coordinates. For point $A(2,9)$ and $A'(\frac{4}{3},6)$, using $y' = ky$, where $y = 9$ and $y' = 6$. Then $6=k\times9$, and $k=\frac{6}{9}=\frac{2}{3}$.

Answer:

$\frac{2}{3}$