Question

A quadrilateral has the following vertices: (4,4) , (6,6) , (8,4) , (6,2) . After dilation, the new quadrilateral has the following vertices: (3,3) , (4.5,4.5) , (6,3) , (4.5,1.5) . What scale factor was used to dilate the quadrilateral?(1 point)

Answer 1

To find the scale factor used to dilate the quadrilateral, we can compare the corresponding vertices of the original quadrilateral and the dilated quadrilateral.

Let’s denote the original vertices as follows:
- \( A(4, 4) \)
- \( B(6, 6) \)
- \( C(8, 4) \)
- \( D(6, 2) \)

And the new, dilated vertices as:
- \( A'(3, 3) \)
- \( B'(4.5, 4.5) \)
- \( C'(6, 3) \)
- \( D'(4.5, 1.5) \)

We will use the formula for scale factor \( k \):

\[
k = \frac{\text{New Coordinate}}{\text{Original Coordinate}}
\]

Now we can compute the scale factor for each corresponding vertex:

1. For point \( A(4, 4) \) to \( A'(3, 3) \):
\[
k_A = \frac{3}{4} = 0.75
\]

2. For point \( B(6, 6) \) to \( B'(4.5, 4.5) \):
\[
k_B = \frac{4.5}{6} = 0.75
\]

3. For point \( C(8, 4) \) to \( C'(6, 3) \):
\[
k_C = \frac{6}{8} = 0.75
\]

4. For point \( D(6, 2) \) to \( D'(4.5, 1.5) \):
\[
k_D = \frac{4.5}{6} = 0.75
\]

Since all calculated scale factors are equal to \( 0.75 \), the scale factor used to dilate the quadrilateral is

\[
\boxed{0.75}
\]