Question

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

Answer 1

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

The original vertices are:
1. (4, 4)
2. (6, 6)
3. (8, 4)
4. (6, 2)

The dilated vertices are:
1. (3, 3)
2. (4.5, 4.5)
3. (6, 3)
4. (4.5, 1.5)

We will determine the scale factor \( k \) by comparing the corresponding x (or y) coordinates of the original and dilated vertices.

1. For the vertex (4, 4) to (3, 3):
\[
k = \frac{3}{4} = 0.75
\]

2. For the vertex (6, 6) to (4.5, 4.5):
\[
k = \frac{4.5}{6} = 0.75
\]

3. For the vertex (8, 4) to (6, 3):
\[
k = \frac{6}{8} = 0.75
\]

4. For the vertex (6, 2) to (4.5, 1.5):
\[
k = \frac{4.5}{6} = 0.75
\]

In each case, the scale factor is consistently \( k = 0.75 \).

Therefore, the scale factor used for the dilation is \( \boxed{0.75} \).