Question

Line AB
contains point A(−4, 1)
and point B (−1, 3)
. Find the coordinates of A′
and B′
after a dilation with a scale factor of 2 with a center point of dilation at the origin.

A′ (−8, 2)
and B′ (−2, 6)
upper a prime left parenthesis negative 8 comma 2 right parenthesis and upper b prime left parenthesis negative 2 comma 6 right parenthesis

A′ (−8, 2)
and B′ (2, −6)
upper A prime left parenthesis negative 8 comma 2 right parenthesis and upper B prime left parenthesis 2 comma negative 6 right parenthesis

A′ (−5, −2)
and B′ (−2, 6)
upper A prime left parenthesis negative 5 comma negative 2 right parenthesis and upper B prime left parenthesis negative 2 comma 6 right parenthesis

A′ (8, −2)
and B′ (2, −6)

Answer 1

To find the coordinates of points A' and B' after a dilation with a scale factor of 2 centered at the origin, we apply the dilation formula:

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

where \( (x, y) \) are the original points, \( (x', y') \) are the new points after dilation, and \( k \) is the scale factor.

Given the points:
- Point A: \( A(-4, 1) \)
- Point B: \( B(-1, 3) \)

And the scale factor \( k = 2 \):

1. For point A:
\[
A' = (2 \cdot -4, 2 \cdot 1) = (-8, 2)
\]

2. For point B:
\[
B' = (2 \cdot -1, 2 \cdot 3) = (-2, 6)
\]

Therefore, after dilation, the coordinates are:
- \( A'(-8, 2) \)
- \( B'(-2, 6) \)

Thus, the correct answer is:

**A′ (−8, 2) and B′ (−2, 6)**