To determine the vertices of the polygon [tex]\( A' B' C' D' \)[/tex] after dilation with a scale factor of [tex]\( \frac{3}{4} \)[/tex], we follow these steps:
1. Identify the initial vertices of the polygon [tex]\( ABCD \)[/tex]:
- [tex]\( A(-4, 6) \)[/tex]
- [tex]\( B(-2, 2) \)[/tex]
- [tex]\( C(4, -2) \)[/tex]
- [tex]\( D(4, 4) \)[/tex]
2. Apply the scale factor [tex]\( \frac{3}{4} \)[/tex] to each coordinate.
To dilate a point [tex]\( (x, y) \)[/tex] by a scale factor [tex]\( k \)[/tex], the new coordinates [tex]\( (x', y') \)[/tex] are given by [tex]\( x' = kx \)[/tex] and [tex]\( y' = ky \)[/tex].
Calculating each vertex:
- Vertex [tex]\( A \)[/tex]:
- Original coordinates: [tex]\( (-4, 6) \)[/tex]
- New coordinates: [tex]\( \left( -\frac{3}{4} \times 4, \frac{3}{4} \times 6 \right) = (-3, 4.5) \)[/tex]
- Vertex [tex]\( B \)[/tex]:
- Original coordinates: [tex]\( (-2, 2) \)[/tex]
- New coordinates: [tex]\( \left( -\frac{3}{4} \times 2, \frac{3}{4} \times 2 \right) = (-1.5, 1.5) \)[/tex]
- Vertex [tex]\( C \)[/tex]:
- Original coordinates: [tex]\( (4, -2) \)[/tex]
- New coordinates: [tex]\( \left( \frac{3}{4} \times 4, \frac{3}{4} \times -2 \right) = (3, -1.5) \)[/tex]
- Vertex [tex]\( D \)[/tex]:
- Original coordinates: [tex]\( (4, 4) \)[/tex]
- New coordinates: [tex]\( \left( \frac{3}{4} \times 4, \frac{3}{4} \times 4 \right) = (3, 3) \)[/tex]
3. Compile the new vertices:
So, the vertices of the dilated polygon [tex]\( A' B' C' D' \)[/tex] are:
[tex]\[ A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), D'(3, 3) \][/tex]
Therefore, the correct answer is:
[tex]\[ A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), D'(3, 3) \][/tex]
This matches the first option provided:
[tex]\[ A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), D'(3, 3) \][/tex]
