Question

y
7-
6-
B
5-
A
4-
3-
2-
-1
-0
x
-2 -1.60
1
2
3
4
5
6
7
8
4-1-
A dilation of AB¯¯¯¯¯¯¯¯
occurs using a scale factor of 12
and a center of dilation at the origin. Prove that A′B′¯¯¯¯¯¯¯¯¯¯
is half the length of AB¯¯¯¯¯¯¯¯
by finding the points A′
and B′
.

(1 point)
Responses

A′
is at (4,8)
and B′
is at (12,12)
.
upper A prime is at left parenthesis 4 comma 8 right parenthesis and upper B prime is at left parenthesis 12 comma 12 right parenthesis .

A′
is at (1,2)
and B′
is at (3,3)
.
upper A prime is at left parenthesis 1 comma 2 right parenthesis and upper B prime is at left parenthesis 3 comma 3 right parenthesis .

A′
is at (−1,−2)
and B′
is at (−3,−3)
.
upper A prime is at left parenthesis negative 1 comma negative 2 right parenthesis and upper B prime is at left parenthesis negative 3 comma negative 3 right parenthesis .

A′
is at (1,5)
and B′
is at (3,6)
.
upper A prime is at left parenthesis 1 comma 5 right parenthesis and upper B prime is at left parenthesis 3 comma 6 right parenthesis .

Answer 1

To find points \( A' \) and \( B' \) after a dilation of segment \( AB \) with a scale factor of \( \frac{1}{2} \) and a center of dilation at the origin, we first need to establish the coordinates of points \( A \) and \( B \).

Let’s assume the original coordinates of points \( A \) and \( B \) are as follows (for the sake of this example):

- Let \( A = (x_1, y_1) \)
- Let \( B = (x_2, y_2) \)

The formula for dilation from the origin (0,0) with a scale factor \( k \) is:

\[
(x', y') = (kx, ky)
\]

Given that the scale factor is \( \frac{1}{2} \), the coordinates of \( A' \) and \( B' \) will be:

\[
A' = \left(\frac{1}{2}x_1, \frac{1}{2}y_1\right)
\]
\[
B' = \left(\frac{1}{2}x_2, \frac{1}{2}y_2\right)
\]

Now, to check which of the provided responses matches with this dilation.

1. If we determine the distance between \( A \) and \( B \):

\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

2. The length of \( A'B' \):

\[
A'B' = \sqrt{\left(\frac{1}{2}(x_2 - x_1)\right)^2 + \left(\frac{1}{2}(y_2 - y_1)\right)^2} = \frac{1}{2}AB
\]

Thus, we need to check the result against the options provided:

**Given Dilation Results:**

- \( A' \) is at \( (4,8) \) and \( B' \) at \( (12,12) \)
- \( A' \) is at \( (1,2) \) and \( B' \) at \( (3,3) \)
- \( A' \) is at \( (-1,-2) \) and \( B' \) at \( (-3,-3) \)
- \( A' \) is at \( (1,5) \) and \( B' \) at \( (3,6) \)

### Determine Every Option:
1. If \( A = (8, 16) \) and \( B = (24, 24) \), then the coordinates are correct for \( A' = (4,8) \) and \( B' = (12,12) \) which would not be half.
2. If \( A = (2, 4) \) and \( B = (6, 6) \), this results in \( A' = (1,2) \) and \( B' = (3,3) \).
3. \( A' = (-1,-2) \) and \( B' = (-3,-3) \) corresponds to \( A = (-2,-4) \) and \( B = (-6,-6)\).
4. \( A' = (1,5) \) and \( B' = (3,6) \), implying original points \( A = (2,10) \) and \( B = (6,12)\).

From analyzing, the option that accurately represents a dilation resulting in half the original length while having a center of dilation at the origin is when points are transformed to:

- **A′ is at (1,2) and B′ is at (3,3).**

Thus, the answer is:

**A′ is at (1,2) and B′ is at (3,3).**