To determine the coordinates of the dilated triangle [tex]\( A' B' C' \)[/tex] with the original triangle [tex]\( A B C \)[/tex] located at points [tex]\( A = (-6, 4) \)[/tex], [tex]\( B = (2, 4) \)[/tex], and [tex]\( C = (1, -2) \)[/tex], and given that the dilation is from the origin by a scale factor of 0.5, we should follow these steps:
1. Dilate point A:
- Original coordinates of [tex]\( A \)[/tex]: [tex]\( (-6, 4) \)[/tex]
- Scale factor: 0.5
- New coordinates of [tex]\( A' \)[/tex]: Multiply each coordinate of [tex]\( A \)[/tex] by the scale factor [tex]\( 0.5 \)[/tex]:
[tex]\[
A' = (-6 \times 0.5, 4 \times 0.5) = (-3, 2)
\][/tex]
2. Dilate point B:
- Original coordinates of [tex]\( B \)[/tex]: [tex]\( (2, 4) \)[/tex]
- Scale factor: 0.5
- New coordinates of [tex]\( B' \)[/tex]: Multiply each coordinate of [tex]\( B \)[/tex] by the scale factor [tex]\( 0.5 \)[/tex]:
[tex]\[
B' = (2 \times 0.5, 4 \times 0.5) = (1, 2)
\][/tex]
3. Dilate point C:
- Original coordinates of [tex]\( C \)[/tex]: [tex]\( (1, -2) \)[/tex]
- Scale factor: 0.5
- New coordinates of [tex]\( C' \)[/tex]: Multiply each coordinate of [tex]\( C \)[/tex] by the scale factor [tex]\( 0.5 \)[/tex]:
[tex]\[
C' = (1 \times 0.5, -2 \times 0.5) = (0.5, -1)
\][/tex]
So, the coordinates of the dilated triangle [tex]\( A' B' C' \)[/tex] are:
[tex]\[
A' = (-3.0, 2.0), \quad B' = (1.0, 2.0), \quad C' = (0.5, -1.0)
\][/tex]
These calculations give us the new positions of the vertices after dilation.
