Question

Determine the scale factor used in the dilation of line segment AB
, with a center point of dilation at the origin
shape one A (9,9) B (6,6)
shape two A (3,3) B (2,2)

Answer 1

To determine the scale factor used in the dilation of line segment \( AB \) with a center point of dilation at the origin, we first need to find the distance of points A and B from the origin (0,0) for both shapes.

### Shape One:
- Point A (9, 9)
- Point B (6, 6)

Calculating the distances from the origin:
\[
\text{Distance from origin to A} = \sqrt{(9 - 0)^2 + (9 - 0)^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2}
\]
\[
\text{Distance from origin to B} = \sqrt{(6 - 0)^2 + (6 - 0)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
\]

### Shape Two:
- Point A (3, 3)
- Point B (2, 2)

Calculating the distances from the origin:
\[
\text{Distance from origin to A} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
\[
\text{Distance from origin to B} = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

### Finding the Scale Factor:
Now, we need to calculate the scale factor based on the distances from the origin:

- For point A:
\[
\text{Scale Factor for A} = \frac{\text{Distance from origin to A in Shape 2}}{\text{Distance from origin to A in Shape 1}} = \frac{3\sqrt{2}}{9\sqrt{2}} = \frac{3}{9} = \frac{1}{3}
\]

- For point B:
\[
\text{Scale Factor for B} = \frac{\text{Distance from origin to B in Shape 2}}{\text{Distance from origin to B in Shape 1}} = \frac{2\sqrt{2}}{6\sqrt{2}} = \frac{2}{6} = \frac{1}{3}
\]

Since the scale factor is consistent for both points, the scale factor used in the dilation of the line segment \( AB \) is:

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