Question

Triangle ABC A B C has vertices point A (−2,4) A ( − 2 , 4 ) , point B (−6,8) B ( − 6 , 8 ) , and point C (−2,4) C ( − 2 , 4 ) . After dilation, triangle A′B′C′ A ′ B ′ C ′ has vertices point A′ (−1,2) A ′ ( − 1 , 2 ) , point B′ (−3,4) B ′ ( − 3 , 4 ) , and point C′ (−1,2) C ′ ( − 1 , 2 ) . Determine the scale factor for the dilation of triangle ABC A B C with a center point of dilation at the origin (0,0) ( 0 , 0 ) . (1 point) Responses 12 Start Fraction 1 over 2 End Fraction −12 negative Start Fraction 1 over 2 End Fraction −2 negative 2 2

Answer 1

To find the scale factor for the dilation of triangle ABC to triangle A'B'C', we can use the coordinates of corresponding points.

1. **Identify the corresponding points**:
- A (−2, 4) dilates to A' (−1, 2)
- B (−6, 8) dilates to B' (−3, 4)
- C (−2, 4) dilates to C' (−1, 2) (C and A are also the same point)

2. **Calculate the scale factor**:
The scale factor \( k \) can be found by comparing the coordinates of a corresponding point before and after dilation.

Using point A and A':

\[
k = \frac{x'}{x} = \frac{-1}{-2} = \frac{1}{2}
\]
\[
k = \frac{y'}{y} = \frac{2}{4} = \frac{1}{2}
\]

Using point B and B':

\[
k = \frac{x'}{x} = \frac{-3}{-6} = \frac{1}{2}
\]
\[
k = \frac{y'}{y} = \frac{4}{8} = \frac{1}{2}
\]

3. **Conclusion**:
Since both pairs of corresponding points yield the same scale factor, we conclude that the scale factor for the dilation is:

\[
\frac{1}{2}
\]

Thus, the correct answer is **Start Fraction 1 over 2 End Fraction**.