Question

analyzing a dilation
derek tried to dilate the isosceles trapezoid abdc with the center of dilation at the origin with a scale factor of 2. check his work and complete the statements.
the slope of ac is
the slope of eg is
the polygons are not dilations of each other because

Explanation:

Step1: Find coordinates of A, C, E, G

From the graph: \( A(-2, 4) \), \( C(-3, 0) \), \( E(-1, 2) \), \( G(-2, 0) \).

Step2: Calculate slope of AC

Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
For \( AC \): \( m_{AC} = \frac{4 - 0}{-2 - (-3)} = \frac{4}{1} = 4 \).

Step3: Calculate slope of EG

For \( EG \): \( m_{EG} = \frac{2 - 0}{-1 - (-2)} = \frac{2}{1} = 2 \).

Step4: Analyze dilation

Dilation preserves slope (parallel lines remain parallel), but here \( m_{AC}
eq m_{EG} \), so sides aren’t parallel. Also, dilation requires proportional distances from the center (origin). Check distances:
\( OA = \sqrt{(-2)^2 + 4^2} = \sqrt{20} \), \( OE = \sqrt{(-1)^2 + 2^2} = \sqrt{5} \). \( \frac{OE}{OA} = \frac{\sqrt{5}}{\sqrt{20}} = \frac{1}{2} \), but for \( OC = 3 \), \( OG = 2 \), \( \frac{OG}{OC} = \frac{2}{3}
eq \frac{1}{2} \). So scale factor isn’t consistent (or slopes differ), so not a dilation.

Answer:

s:
The slope of \( AC \) is \( \boldsymbol{4} \).
The slope of \( EG \) is \( \boldsymbol{2} \).
The polygons are not dilations of each other because the slopes of corresponding sides are not equal (or the scale factor is not consistent for all corresponding points from the center of dilation).