Question

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 dropdown. the slope of eg is dropdown. the polygons are not dilations of each other because dropdown. (image: coordinate plane with trapezoids abdc, efgh, points a, b, c, d, e, f, g, h plotted on grid with x, y axes)

Explanation:

To solve this, we first identify coordinates from the graph. Let's assume:

• Point \( A(-2, 2) \), \( C(-4, 0) \), \( E(-1, 1) \), \( G(-2, 0) \).

Step 1: Slope of \( AC \)

Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
For \( A(-2, 2) \) and \( C(-4, 0) \):
\( m_{AC} = \frac{0 - 2}{-4 - (-2)} = \frac{-2}{-2} = 1 \)

Step 2: Slope of \( EG \)

For \( E(-1, 1) \) and \( G(-2, 0) \):
\( m_{EG} = \frac{0 - 1}{-2 - (-1)} = \frac{-1}{-1} = 1 \)

Step 3: Why not dilations?

Dilations preserve slope (parallel lines remain parallel) but also require corresponding sides to be proportional. Here, slopes are equal, but check side lengths:

• Length \( AC \): \( \sqrt{(-4 + 2)^2 + (0 - 2)^2} = \sqrt{8} = 2\sqrt{2} \)
• Length \( EG \): \( \sqrt{(-2 + 1)^2 + (0 - 1)^2} = \sqrt{2} \)

Scale factor should be \( \frac{EG}{AC} = \frac{\sqrt{2}}{2\sqrt{2}} = \frac{1}{2} \), but Derek used scale factor 2. Or, check center of dilation: lines from origin to corresponding vertices should be colinear. For \( A(-2,2) \) and \( E(-1,1) \), the line \( y = -x \) (wait, no: \( E \) is \( (-1,1) \), \( A \) is \( (-2,2) \)—so \( E \) is \( \frac{1}{2}A \), meaning scale factor \( \frac{1}{2} \), not 2. Thus, corresponding sides are not in the ratio of the scale factor (2), or the center of dilation is not respected (lines from origin to \( A \) and \( E \) are colinear, but for \( C(-4,0) \) and \( G(-2,0) \), \( G \) is \( \frac{1}{2}C \), so scale factor \( \frac{1}{2} \), not 2. So polygons are not dilations because the scale factor is inconsistent (or Derek’s scale factor 2 is wrong, as the actual ratio is \( \frac{1}{2} \)).

Answer:

s:

• Slope of \( AC \): \( \boldsymbol{1} \)
• Slope of \( EG \): \( \boldsymbol{1} \)
• Reason: The corresponding sides are not in the ratio of the scale factor (2) (or the scale factor used is incorrect, as the actual ratio is \( \frac{1}{2} \)).