Question

Select the coordinates A′
and B′
after dilation of the line segment AB
with a scale factor of 1/3
, centered at the origin where the endpoints are A(−6,3)
and B(−12,9)
.(1 point)
Responses

A′(−2,1)
and B′(−12,9)
upper A prime left parenthesis negative 2 comma 1 right parenthesis and upper B prime left parenthesis negative 12 comma 9 right parenthesis

A′(−18,9)
and B′(−36,27)
upper A prime left parenthesis negative 18 comma 9 right parenthesis and upper B prime left parenthesis negative 36 comma 27 right parenthesis

A′(−2,1)
and B′(−4,3)
upper A prime left parenthesis negative 2 comma 1 right parenthesis and upper B prime left parenthesis negative 4 comma 3 right parenthesis

A′(−6,3)
and B′(−4,3)
upper A prime left parenthesis negative 6 comma 3 right parenthesis and upper B prime left parenthesis negative 4 comma 3 right parenthesis

Answer 1

To find the coordinates of the points A′ and B′ after dilation of the line segment AB with a scale factor of \( \frac{1}{3} \), centered at the origin, we will apply the dilation transformation to each point.

The formula for dilation centered at the origin is:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

where \( k \) is the scale factor, and \( (x, y) \) are the original coordinates.

The endpoints provided are:
- A(−6, 3)
- B(−12, 9)

Applying the dilation with scale factor \( k = \frac{1}{3} \):

1. For point A(−6, 3):

\[
A' = \left(\frac{1}{3} \cdot -6, \frac{1}{3} \cdot 3\right) = \left(-2, 1\right)
\]

2. For point B(−12, 9):

\[
B' = \left(\frac{1}{3} \cdot -12, \frac{1}{3} \cdot 9\right) = \left(-4, 3\right)
\]

Thus, the coordinates after dilation are:

\[
A'(-2, 1) \quad \text{and} \quad B'(-4, 3)
\]

The correct response is:

**A′(−2,1) and B′(−4,3)**.