Question

Determine the scale factor used in the dilation of line segment AB, centered at the origin. the scale factor is

Answer 1

To determine the scale factor used in the dilation of a line segment \(AB\) centered at the origin, you'll generally follow these steps:

1. **Identify the coordinates of points A and B:** Let's say \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the coordinates of the endpoints of the line segment.

2. **Understand dilation:** A dilation centered at the origin with a scale factor \(k\) will transform points according to the following formula:

\[
A'(x_1', y_1') = (k \cdot x_1, k \cdot y_1)
\]
\[
B'(x_2', y_2') = (k \cdot x_2, k \cdot y_2)
\]

where \(A'\) and \(B'\) are the new points after dilation.

3. **Determine the scale factor:** If you are given the new coordinates of points \(A'\) and \(B'\), you can determine the scale factor \(k\) using the coordinates of the original points \(A\) and \(B\). The scale factor can be found using the formulas:

\[
k = \frac{x_1'}{x_1} = \frac{y_1'}{y_1} = \frac{x_2'}{x_2} = \frac{y_2'}{y_2}
\]

Make sure the original coordinates are nonzero, as a dilation by zero wouldn't be meaningful.

**Example:** If \(A(2, 3)\) and \(B(4, 6)\) are dilated to \(A'(4, 6)\) and \(B'(8, 12)\), the scale factor \(k\) can be calculated as:

\[
k = \frac{x_1'}{x_1} = \frac{4}{2} = 2 \quad \text{and} \quad k = \frac{y_1'}{y_1} = \frac{6}{3} = 2
\]

The same will apply for point B.

Thus, the scale factor in this example would be \(2\).

If you provide specific coordinates for points \(A\) and \(B\) and their dilated images, I can help calculate the exact scale factor.