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:
- \( A(4, 4) \)
- \( B(6, 6) \)
- \( C(8, 4) \)
- \( D(6, 2) \)
The dilated vertices are:
- \( A'(3, 3) \)
- \( B'(4.5, 4.5) \)
- \( C'(6, 3) \)
- \( D'(4.5, 1.5) \)
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:
\[ \boxed{\frac{3}{4}} \]