Question

find the scale factor of the line segment dilation.
\\(overline{ab}\\): endpoints \\((-1, -2)\\) and \\((-2, -4)\\) to \\(overline{ab}\\): endpoints at \\((-2, -4)\\) and \\((-4, -8)\\).
a \\(-\frac{1}{2}\\)
b \\(\frac{1}{2}\\)
c \\(2\\)
d \\(-2\\)

Explanation:

Step1: Identify corresponding points

Let \( A(-1, -2) \), \( B(-2, -4) \), \( A'(-2, -4) \), \( B'(-4, -8) \). So \( A \) corresponds to \( A' \), \( B \) corresponds to \( B' \).

Step2: Use dilation formula for coordinates

The dilation of a point \( (x, y) \) with scale factor \( k \) is \( (kx, ky) \). For point \( A(-1, -2) \) to \( A'(-2, -4) \):
Check \( k \times (-1)= -2 \) and \( k \times (-2)= -4 \). Solving \( k \times (-1)= -2 \), we get \( k = \frac{-2}{-1}=2 \).
Check with \( B(-2, -4) \) to \( B'(-4, -8) \): \( k \times (-2)= -4 \) gives \( k = \frac{-4}{-2}=2 \), and \( k \times (-4)= -8 \) gives \( k = \frac{-8}{-4}=2 \).

Answer:

C. 2