Question

What is the scale factor of a triangle with a vertex of A(–6, 4) that has been dilated with a center of dilation at the origin so the vertex of its image is A prime(–24, 16)?

Answer 1

To determine the scale factor of a dilation from the origin, you can use the coordinates of the original point and its dilated image.

Given:
- Original point \( A(-6, 4) \)
- Dilated image point \( A'(-24, 16) \)

The scale factor \( k \) can be determined using the formula:

\[
k = \frac{\text{coordinate of the image}}{\text{coordinate of the original point}}
\]

We can find \( k \) for both the x and y coordinates:

1. For the x-coordinate:
\[
k_x = \frac{-24}{-6} = \frac{24}{6} = 4
\]

2. For the y-coordinate:
\[
k_y = \frac{16}{4} = 4
\]

Since both scale factors \( k_x \) and \( k_y \) are equal, the overall scale factor \( k \) is:

\[
k = 4
\]

Thus, the scale factor of the triangle is \( \boxed{4} \).