Question

Determine the scale factor used in the dilation of line segment AB, with a center point of dilation at the origin.

Original: A = (9, 9) B = (6, 6)

After dilation: A = (3, 3) B = (2, 2)

Answer 1

To find the scale factor used in the dilation of the line segment AB from its original coordinates to its new coordinates, we can use the formula for dilation.

Given:
- Original Points:
- \( A = (9, 9) \)
- \( B = (6, 6) \)

- Dilation Points:
- \( A' = (3, 3) \)
- \( B' = (2, 2) \)

The center of dilation is at the origin \((0, 0)\).

To find the scale factor \( k \), we calculate it using the formula:

\[
k = \frac{\text{distance from origin to the new point}}{\text{distance from origin to the original point}}
\]

Let's calculate \( k \) for point \( A \):

1. The distance from the origin to the original point \( A \):
\[
d_A = \sqrt{(9 - 0)^2 + (9 - 0)^2} = \sqrt{9^2 + 9^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2}
\]

2. The distance from the origin to the new point \( A' \):
\[
d_{A'} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

Now we can find \( k \) for point \( A \):
\[
k = \frac{d_{A'}}{d_A} = \frac{3\sqrt{2}}{9\sqrt{2}} = \frac{3}{9} = \frac{1}{3}
\]

Next, we'll check that the same scale factor applies for point \( B \):

1. The distance from the origin to the original point \( B \):
\[
d_B = \sqrt{(6 - 0)^2 + (6 - 0)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
\]

2. The distance from the origin to the new point \( B' \):
\[
d_{B'} = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

Now we can find \( k \) for point \( B \):
\[
k = \frac{d_{B'}}{d_B} = \frac{2\sqrt{2}}{6\sqrt{2}} = \frac{2}{6} = \frac{1}{3}
\]

Since the scale factor is the same for both points \( A \) and \( B \), the scale factor for the dilation is

\[
\boxed{\frac{1}{3}}
\]