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 for the dilation of the quadrilateral, we can compare the corresponding vertices of the original quadrilateral and the dilated quadrilateral.

The original vertices are:
1. \( A(4, 4) \)
2. \( B(6, 6) \)
3. \( C(8, 4) \)
4. \( D(6, 2) \)

The dilated vertices are:
1. \( A'(3, 3) \)
2. \( B'(4.5, 4.5) \)
3. \( C'(6, 3) \)
4. \( D'(4.5, 1.5) \)

We can use any pair of corresponding vertices to find the scale factor. Let's use points A and A':

The formula for dilation, if the scale factor is \( k \), is:
\[
A' = (k \cdot x_A, k \cdot y_A)
\]

For point \( A \):
\[
A' = (3, 3), \quad A = (4, 4)
\]

Setting up the equations for the scale factor \( k \):
\[
3 = k \cdot 4 \quad (1)
\]
\[
3 = k \cdot 4 \quad (2)
\]

From equation (1):
\[
k = \frac{3}{4}
\]

Now, let's check this scale factor with point B and B':

For point \( B \):
\[
B' = (4.5, 4.5), \quad B = (6, 6)
\]

Setting up the equations again for the scale factor \( k \):
\[
4.5 = k \cdot 6 \quad (3)
\]
\[
4.5 = k \cdot 6 \quad (4)
\]

From equation (3):
\[
k = \frac{4.5}{6} = \frac{3}{4}
\]

We'll check with point C to see if it remains consistent:

For point \( C \):
\[
C' = (6, 3), \quad C = (8, 4)
\]

Setting up the equations:
\[
6 = k \cdot 8 \quad (5)
\]
\[
3 = k \cdot 4 \quad (6)
\]

From equation (5):
\[
k = \frac{6}{8} = \frac{3}{4}
\]

Finally, checking with point D:

For point \( D \):
\[
D' = (4.5, 1.5), \quad D = (6, 2)
\]

Setting up the equations:
\[
4.5 = k \cdot 6 \quad (7)
\]
\[
1.5 = k \cdot 2 \quad (8)
\]

From equation (7):
\[
k = \frac{4.5}{6} = \frac{3}{4}
\]
From equation (8):
\[
k = \frac{1.5}{2} = \frac{3}{4}
\]

All pairs of corresponding vertices give the same scale factor. Therefore, the scale factor used to dilate the quadrilateral is:

\[
\boxed{\frac{3}{4}}
\]