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) what were the original coordinates?

Answer 1

To find the dilated coordinates \( A' \) and \( B' \) of the line segment \( AB \) after applying a dilation with a scale factor of \( \frac{1}{3} \) centered at the origin, we can use the dilation formula:

\[
A' = (k \cdot x_A, k \cdot y_A)
\]
\[
B' = (k \cdot x_B, k \cdot y_B)
\]

where \( k \) is the scale factor, and \( (x_A, y_A) \) and \( (x_B, y_B) \) are the coordinates of points \( A \) and \( B \).

Given:
- \( A(-6, 3) \)
- \( B(-12, 9) \)
- Scale factor \( k = \frac{1}{3} \)

Now, applying the formula:

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

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

Thus, the dilated coordinates after applying the scale factor are:
- \( A'(-2, 1) \)
- \( B'(-4, 3) \)

The original coordinates are:
- \( A(-6, 3) \)
- \( B(-12, 9) \)