Question

Line AB
and line CD
are parallel lines. Line AB
contains the points A(−5,−5)
and B(−3,−2)
, and line CD
contains the points C(−7,−2,)
and D(−5,1)
. Describe the effect of the dilation with a scale factor of 3 and a center point of dilation at the origin (0,0)
.

Option #1: After dilation, the lines are three units farther from the point of dilation. The resulting points are A′ (−2,−2)
, B′ (0,1)
, C′(−4,1)
, and D′(−2,4)
. The dilated lines are parallel.

Option #2: After dilation, the lines are three units closer to the point of dilation. The resulting points are A′ (−53,−53)
, B′(−1,−23)
, C′(−72,−23)
, and D′(−53,13)
. The dilated lines are parallel.

Option #3: After dilation, the lines are three units farther from the point of dilation. The resulting points are A′(−15,−15)
, B′(−9,−6)
, C′(−21,−6)
, and D′(−15,3)
. The dilated lines are parallel.

(1 point)
Option #

Answer 1

To determine the effect of the dilation with a scale factor of 3 centered at the origin (0, 0), we multiply the coordinates of each point by 3.

1. **Calculating the dilated points:**

- For point A(-5, -5):
- A' = (3 * -5, 3 * -5) = (-15, -15)

- For point B(-3, -2):
- B' = (3 * -3, 3 * -2) = (-9, -6)

- For point C(-7, -2):
- C' = (3 * -7, 3 * -2) = (-21, -6)

- For point D(-5, 1):
- D' = (3 * -5, 3 * 1) = (-15, 3)

2. **Checking the results:**
The coordinates for the dilated points are:
- A' = (-15, -15)
- B' = (-9, -6)
- C' = (-21, -6)
- D' = (-15, 3)

Next, we check the options provided:

- **Option #1**: A' (−2,−2), B' (0,1), C' (−4,1), D' (−2,4)
- **Option #2**: A' (−53,−53), B' (−1,−23), C' (−72,−23), D' (−53,13)
- **Option #3**: A' (−15,−15), B' (−9,−6), C' (−21,−6), D' (−15,3)

The correct option based on our calculations is **Option #3**:

After dilation, the lines are three units farther from the point of dilation. The resulting points are A' (−15,−15), B' (−9,−6), C' (−21,−6), and D' (−15,3). The dilated lines are parallel.