Question

Angle DEF
contains the points D(4,4)
, E(8,0)
, and F(8,8)
. Describe the effect of the dilation of the angle with a scale factor of 14
and a center point of dilation at the origin (0,0)
. Enter the option number of your response.

Option #1: After dilation, the angle is four times farther from the point of dilation. The resulting points are A′(16,16)
, B′(32,0)
, and C′(32,32)
. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle.

Option #2: After dilation, the angle is four times closer to the point of dilation. The resulting points are A′(1,1)
, B′(2,0)
, and C′(2,2)
. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle.

Option #3: After dilation, the angle is four times closer to the point of dilation. The resulting points are A′(174,174)
, B′(334,14)
, and C′(334,334)
. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle.

(1 point)
Option #

Answer 1

To analyze the dilation of the points D(4,4), E(8,0), and F(8,8) with a scale factor of 14 and a center of dilation at the origin (0,0), we apply the following formula for dilation:

If \((x, y)\) is a point and the center of dilation is at \((0, 0)\), the new point after dilation will be:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
where \(k\) is the scale factor.

For each point:

1. **Point D(4,4)**:
\[
D' = (14 \cdot 4, 14 \cdot 4) = (56, 56)
\]

2. **Point E(8,0)**:
\[
E' = (14 \cdot 8, 14 \cdot 0) = (112, 0)
\]

3. **Point F(8,8)**:
\[
F' = (14 \cdot 8, 14 \cdot 8) = (112, 112)
\]

After dilation, we have the new points:
- \(D'(56,56)\)
- \(E'(112,0)\)
- \(F'(112,112)\)

The dilation moves the points further away from the origin by a factor of 14. The angle measurement remains the same, and the corresponding rays (lines) between the original and dilated angles will remain parallel.

Now evaluating the options:
- **Option #1** is incorrect since it states the new points are \(A'(16,16)\), \(B'(32,0)\), and \(C'(32,32)\), which do not match our calculations.
- **Option #2** is incorrect because it states the new points are much closer to the origin than the original (and also incorrect points).
- **Option #3** is also incorrect, as it again states incorrect points.

None of the provided options correctly describe the dilation of angle DEF to the points calculated. However, since Option #1 describes the angle being "four times farther" from the point of dilation, it is the one that aligns most closely with the understanding of dilation moving the points further away, albeit with incorrect point values.

**Final response**: Option #1.