Question

line segment ab. a is (2,3) to (6,9) and b is (4,1) to (12,3) what is the dilation

Answer 1

To determine the dilation between the two line segments \( AB \) and \( CD \) where:

- Line segment \( AB \) has endpoints \( A(2, 3) \) and \( B(6, 9) \).
- Line segment \( CD \) has endpoints \( C(4, 1) \) and \( D(12, 3) \).

### Step 1: Calculate the lengths of the line segments

**Length of segment \( AB \)**:
Using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

For \( AB \) (from \( A(2, 3) \) to \( B(6, 9) \)):

\[
AB = \sqrt{(6 - 2)^2 + (9 - 3)^2} = \sqrt{(4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]

**Length of segment \( CD \)**:
For \( CD \) (from \( C(4, 1) \) to \( D(12, 3) \)):

\[
CD = \sqrt{(12 - 4)^2 + (3 - 1)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17}
\]

### Step 2: Calculate the dilation factor

The dilation factor \( k \) is the ratio of the lengths of the segments:

\[
k = \frac{\text{Length of } AB}{\text{Length of } CD} = \frac{2\sqrt{13}}{2\sqrt{17}} = \frac{\sqrt{13}}{\sqrt{17}}
\]

### Conclusion

The dilation factor \( k \) between the line segments \( AB \) and \( CD \) is:

\[
k = \frac{\sqrt{13}}{\sqrt{17}}
\]

This represents how the first segment \( AB \) dilates to become segment \( CD \).