Question

triangle (abc) has vertices (a(2,4)), (b(0, - 4)), and (c(-1,-2)). it is dilated with a center at ((0,0)) to produce triangle (abc) with vertices (a(12,24)), (b(0,-24)), and (c(-6,-12)). what is the scale factor of the dilation? enter the correct answer in the box.

Explanation:

Step1: Recall dilation formula

For a dilation centered at the origin $(0,0)$, if a point $(x,y)$ is dilated to $(x',y')$, the scale - factor $k$ is given by $x' = kx$ and $y'=ky$. Let's use point $A(2,4)$ which is dilated to $A'(12,24)$.

Step2: Find scale - factor using x - coordinates

We know that $x' = kx$. Substituting $x = 2$ and $x'=12$ into the formula, we get $12=k\times2$. Solving for $k$ gives $k=\frac{12}{2}=6$.

Step3: Check using y - coordinates

We also know that $y' = ky$. Substituting $y = 4$ and $y' = 24$ into the formula, we get $24=k\times4$. Solving for $k$ gives $k=\frac{24}{4}=6$.

Answer:

6