Question

select the correct answer.
a triangle with vertices at ( a(0,0) ), ( b(0,4) ), and ( c(6,0) ) is dilated to yield a triangle with vertices at ( a(0,0) ), ( b(0,10) ), and ( c(15,0) ). the origin is the center of dilation. what is the scale factor of the dilation?
a. 1.5
b. 2
c. 2.5
d. 3

Explanation:

Step1: Recall the formula for scale factor

The scale factor \(k\) of a dilation with center at the origin is given by \(k=\frac{\text{coordinate of image}}{\text{coordinate of pre - image}}\) for non - zero coordinates.

Step2: Calculate the scale factor using \(y\) - coordinates (or \(x\) - coordinates)

We can use the \(y\) - coordinate of point \(B\) and \(B'\). The \(y\) - coordinate of \(B\) is \(y = 4\) and the \(y\) - coordinate of \(B'\) is \(y'=10\).
Using the formula \(k=\frac{y'}{y}\), we substitute \(y = 4\) and \(y'=10\) into the formula: \(k=\frac{10}{4}=\frac{5}{2}=2.5\).
We can also check using the \(x\) - coordinate of \(C\) (\(x = 6\)) and \(C'\) (\(x'=15\)). Using the formula \(k=\frac{x'}{x}\), we substitute \(x = 6\) and \(x'=15\) into the formula: \(k=\frac{15}{6}=\frac{5}{2}=2.5\)

Answer:

C. 2.5