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"The x-axis of a coordinate plane ranges from negative 10 to 12 and the y-axis ranges from negative 11 to 15, both in 2-unit increments. Three points are plotted, one of which serves as the origin for two line segments out of which rays extend. The points are labeled upper D, upper E, and upper F. Upper D is located at left parenthesis negative 6 comma 12 right parenthesis. Upper E is located at left parenthesis 6 comma 0 right parenthesis. Upper F is located at left parenthesis 0 comma negative 6 right parenthesis.

Describe the effect of the dilation of the angle DEF
with a scale factor of 16
and a center point of dilation at the origin (0,0)
.

(1 point)
Responses

After dilation, the angle is six 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′(−36,72)
, E′(36,0)
, and F′(0,−36)
.
After dilation, the angle is six 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 upper D prime left parenthesis negative 36 comma 72 right parenthesis , upper E prime left parenthesis 36 comma 0 right parenthesis , and upper F prime left parenthesis 0 comma negative 36 right parenthesis .

After dilation, the angle is six 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′(−356,736)
, E′(376,16)
, and F′(0,−356)
.
After dilation, the angle is six 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 upper D prime left parenthesis Start Fraction negative 35 over 6 End Fraction comma Start Fraction 73 over 6 End Fraction right parenthesis , upper E prime left parenthesis Start Fraction 37 over 6 End Fraction comma Start Fraction 1 over 6 End Fraction right parenthesis , and upper F prime left parenthesis 0 comma Start Fraction negative 35 over 6 End Fraction right parenthesis .

After dilation, the angle is six 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′(−1,2)
, E′(1,0)
, and F′(0,−1)
.
After dilation, the angle is six 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 upper D prime left parenthesis negative 1 comma 2 right parenthesis , upper E prime left parenthesis 1 comma 0 right parenthesis , and upper F prime left parenthesis 0 comma negative 1 right parenthesis .

After dilation, the angle is six 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′(−1,2)
, E′(1,0)
, and F′(0,−1)
.

1 answer

To determine the correct response regarding the dilation of the angle DEF with respect to the origin (0,0) and a scale factor of 16, let's perform the necessary calculations for each point after dilation.

The dilation of a point (x, y) with a scale factor k and center of dilation (0, 0) is calculated as:

  • \( D' = (kx, ky) \)

Applying this formula to each point:

  1. For point D (-6, 12):

    • \( D' = (16 \cdot -6, 16 \cdot 12) = (-96, 192) \)

  2. For point E (6, 0):

    • \( E' = (16 \cdot 6, 16 \cdot 0) = (96, 0) \)

  3. For point F (0, -6):

    • \( F' = (16 \cdot 0, 16 \cdot -6) = (0, -96) \)

The dilated points are:

  • D'(-96, 192)
  • E'(96, 0)
  • F'(0, -96)

Now, examining the provided responses:

  • The resulting points of D'(-36, 72), E'(36, 0), and F'(0, -36) do not match our calculated results, and the explanation about "six times closer" is inaccurate because the scale factor is actually increasing the distance from the origin.

Since a scale factor of 16 makes the points move farther from the origin, the response mentioning "six times closer" does not apply in this case.

Due to these observations, none of the given multiple-choice options correctly describe the dilation process with the scale factor of 16 and should technically reflect that the points move farther away from the origin.

Therefore, the correct conclusion from the information should state that the points have been moved farther away from the origin, but that specific phrasing doesn't appear. It seems that no responses are correct based on the calculations provided.