Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar.

polygon abcd, shown in the figure, is dilated by a scale factor of 8 with the origin as the center of dilation, resulting in the image abcd.

the slope of \overline{cd} is .

Explanation:

Identify coordinates of points C and D

We locate points \(C\) and \(D\) on the coordinate grid.

• Point \(C\) is at \((5, 4)\).
• Point \(D\) is at \((4, 2)\).

Calculate the slope of segment CD

We use the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Substituting the coordinates of \(C(5, 4)\) and \(D(4, 2)\):

$$ m_{CD} = \frac{4 - 2}{5 - 4} = \frac{2}{1} = 2 $$

Determine the slope of the dilated segment

A dilation centered at the origin maps any line segment to a parallel line segment.
Parallel lines have equal slopes.
Therefore, the slope of the image segment \(\overline{C'D'}\) is equal to the slope of the pre-image segment \(\overline{CD}\).

$$ m_{C'D'} = m_{CD} = 2 $$

Answer:

Polygon ABCD, shown in the figure, is dilated by a scale factor of 8 with the origin as the center of dilation, resulting in the image A'B'C'D'.

The slope of \(\overline{C'D'}\) is <blank>2</blank>.