Question

1. a triangle has vertices x(1,1), y(5,1), z(3,4). after a dilation, its image has vertices x(3,3), y(15,3), z(9,12). what is the scale factor?

a. 1/2
b. 2
c. 3
d. 1/3

Explanation:

Step1: Recall dilation formula

For a dilation with center at the origin, if a point $(x,y)$ is dilated to $(x',y')$, the scale - factor $k$ is given by $x'=k\times x$ and $y' = k\times y$. We can use any of the corresponding coordinates of the pre - image and image points. Let's use the $x$ - coordinates of point $X$ and $X'$.
For point $X(1,1)$ and $X'(3,3)$, we use the formula for the $x$ - coordinate: $x'=k\times x$. Here, $x = 1$ and $x'=3$.

Step2: Solve for scale factor

Substitute $x = 1$ and $x'=3$ into the formula $x'=k\times x$. We get $3=k\times1$. So, $k = 3$.
We can double - check using another point. For point $Y(5,1)$ and $Y'(15,3)$. Using the $x$ - coordinates, $x'=k\times x$, where $x = 5$ and $x'=15$. Then $15=k\times5$, and solving for $k$ gives $k=\frac{15}{5}=3$.

Answer:

c. 3