Question

a triangle with vertices at a(20, -30), b(10, -15), and c(5, -20) has been dilated with a center of dilation at the origin. the image of b, point b, has the coordinates (2, -3). what is the scale factor of the dilation?
options:
\\(\frac{1}{10}\\)
\\(\frac{1}{5}\\)
5
10

Explanation:

Step1: Recall dilation formula

For dilation at origin: $(x', y') = k(x, y)$, where $k$ = scale factor, $(x,y)$ = original point, $(x',y')$ = image point.

Step2: Substitute point B and B'

Original B: $(10, -15)$, Image B': $(2, -3)$. Use x-coordinate: $2 = k \times 10$

Step3: Solve for k

$k = \frac{2}{10} = \frac{1}{5}$
Verify with y-coordinate: $-3 = k \times (-15) \implies k = \frac{-3}{-15} = \frac{1}{5}$

Answer:

$\frac{1}{5}$