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"The x-axis of a coordinate plane ranges from negative 7 to 12 and the y-axis ranges from negative 7 to 6, both in 1-unit increments. Three points are plotted, one of which serves as the origin for two rays. The points are labeled upper D, upper E, and upper F. Upper D is located at left parenthesis negative 2 comma negative 4 right parenthesis. Upper E is located at left parenthesis 2 comma 1 right parenthesis. Upper F is located at left parenthesis 3 comma negative 3 right parenthesis.

Describe the effect of the dilation of the angle DEF
with a scale factor of 4 and a center point of dilation at the origin (0,0)
. Enter the option number of your response.

Option #1: After dilation, the lines are four times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are D′(−8,−16)
, E′(8,4)
, and F′(12,−12)
.

Option #2: After dilation, the lines are four times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are D′(2,0)
, E′(6,5)
, and F′(7,1)
.

Option #3: After dilation, the lines are four times closer to the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are D′(−12,−1)
, E′(12,14)
, and F′(34,−34)
.

(1 point)
Option #

1 answer

To analyze the dilation of angle DEF with respect to the origin (0,0) and a scale factor of 4, we apply the dilation formula for each point, which is \( (x', y') = (kx, ky) \) where \( k \) is the scale factor and \( (x, y) \) are the original coordinates of the points.

Given:

  • Point D: \( (-2, -4) \)
  • Point E: \( (2, 1) \)
  • Point F: \( (3, -3) \)

We can calculate the dilated points as follows:

  1. For point D: \[ D' = (4 \cdot -2, 4 \cdot -4) = (-8, -16) \]

  2. For point E: \[ E' = (4 \cdot 2, 4 \cdot 1) = (8, 4) \]

  3. For point F: \[ F' = (4 \cdot 3, 4 \cdot -3) = (12, -12) \]

After the dilation:

  • D' is at \( (-8, -16) \)
  • E' is at \( (8, 4) \)
  • F' is at \( (12, -12) \)

These points maintain the same angle measurement as angle DEF, and the rays are parallel because dilations preserve angle measurements and the directional relationships of points are scaled consistently.

From the options provided, the correct response that matches these results and the properties of dilation is:

Option #1: After dilation, the lines are four times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are D′(−8,−16), E′(8,4), and F′(12,−12).

Thus, the answer is:

Option #1