Question

Line AB
and line CD
are parallel lines. Line AB
contains the points A(1,4)
and B(2,2)
, and line CD
contains the points C(2,5)
and D(3,3)
. Describe the effect of the dilation with a scale factor of 4 and a center point of dilation at the origin (0,0)
.

Option #1: After dilation, the lines are four units farther from the point of dilation. The resulting points are A′ (5,8)
, B′ (6,6)
, C′(6,9)
, and D′(7,7)
.
Option #2: After dilation, the lines are four units farther from the point of dilation. The resulting points are A′ (4,16)
, B′ (8,8)
, C′(8,20)
, and D′(12,12)
.
Option #3: After dilation, the lines are four units closer to the point of dilation. The resulting points are A′ (14,1)
, B′ (12,12)
, C′(12,54)
, and D′(34,34)
.

(1 point)
Option #

Answer 1

To determine the correct effect of the dilation with a scale factor of 4 and a center of dilation at the origin (0,0), we need to apply the dilation formula:

For a point \((x, y)\), when we apply a dilation with scale factor \(k\) from the origin, the new coordinates \((x', y')\) are given by:
\[
(x', y') = (kx, ky)
\]

Now, let's apply this to each of the points A, B, C, and D.

1. For point A(1, 4):
\[
A' = (4 \cdot 1, 4 \cdot 4) = (4, 16)
\]

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

3. For point C(2, 5):
\[
C' = (4 \cdot 2, 4 \cdot 5) = (8, 20)
\]

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

Now we can summarize the new coordinates of the points after dilation:
- A' = (4, 16)
- B' = (8, 8)
- C' = (8, 20)
- D' = (12, 12)

Looking at the options provided, the correct one is:

**Option #2**: After dilation, the lines are four units farther from the point of dilation. The resulting points are A′ (4,16), B′ (8,8), C′(8,20), and D′(12,12).