To determine the new vertices of side [tex]\( A'C' \)[/tex] after dilating the triangle with vertices [tex]\( A=(-3, -2) \)[/tex], [tex]\( B=(-1, 3) \)[/tex], and [tex]\( C=(2, 1) \)[/tex] by a scale factor of 4, we need to follow these steps:
1. Identify the original coordinates of the vertices:
- [tex]\( A = (-3, -2) \)[/tex]
- [tex]\( B = (-1, 3) \)[/tex]
- [tex]\( C = (2, 1) \)[/tex]
2. Apply the scale factor to each coordinate:
The formula for dilation by a scale factor [tex]\( k \)[/tex] involves multiplying each coordinate by [tex]\( k \)[/tex]. Here, our scale factor is [tex]\( 4 \)[/tex].
3. Calculate the new coordinates for vertex [tex]\( A' \)[/tex]:
[tex]\[
A' = (4 \times -3, 4 \times -2) = (-12, -8)
\][/tex]
4. Calculate the new coordinates for vertex [tex]\( B' \)[/tex]:
As [tex]\( B \)[/tex] is not part of side [tex]\( A'C' \)[/tex], in this question context, we focus on calculating the coordinates of [tex]\( A' \)[/tex] and [tex]\( C' \)[/tex]. However, for completeness:
[tex]\[
B' = (4 \times -1, 4 \times 3) = (-4, 12)
\][/tex]
5. Calculate the new coordinates for vertex [tex]\( C' \)[/tex]:
[tex]\[
C' = (4 \times 2, 4 \times 1) = (8, 4)
\][/tex]
6. Determine the location of side [tex]\( A'C' \)[/tex]:
After dilating the triangle, the new coordinates for the vertices of side [tex]\( A'C' \)[/tex] are:
[tex]\[
A' = (-12, -8)
\][/tex]
[tex]\[
C' = (8, 4)
\][/tex]
Thus, the new vertices of side [tex]\( A'C' \)[/tex] are [tex]\( A' = (-12, -8) \)[/tex] and [tex]\( C' = (8, 4) \)[/tex].
