Question

Use the image to answer the question.

A coordinate plane's axes range from negative 3 to 7 by 1-unit increments. 3 points are plotted and joined by solid lines. The points are labeled upper A, upper B, and upper C.

AC¯¯¯¯¯¯¯¯
is a dilation of AB¯¯¯¯¯¯¯¯
, with point B
as the center of dilation. Determine the scale factor.

(1 point)
The scale factor is
.

Answer 1

what is a example of a scale factore

Answer 2

To determine the scale factor of the dilation from segment \( \overline{AB} \) to segment \( \overline{AC} \) with point \( B \) as the center of dilation, you need to find the lengths of the segments \( \overline{AB} \) and \( \overline{AC} \).

1. **Calculate the length of segment \( \overline{AB} \)**:
Use the coordinates of points \( A \) and \( B \) to find the distance between them. The formula for the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

2. **Calculate the length of segment \( \overline{AC} \)**:
Similarly, calculate the distance between points \( A \) and \( C \) using the same distance formula.

3. **Finding the scale factor \( k \)**:
The scale factor \( k \) can be found by dividing the length of the dilated segment \( \overline{AC} \) by the length of the original segment \( \overline{AB} \):
\[
k = \frac{\text{Length of } \overline{AC}}{\text{Length of } \overline{AB}}
\]

Once you have calculated the lengths from the coordinates of points \( A \), \( B \), and \( C \), substitute the values into the formula for \( k \) to find the scale factor.

Since I cannot see the image you mentioned, I would need the coordinates of points \( A \), \( B \), and \( C \) to assist you further. Please provide those coordinates for a more specific calculation.

Answer 3

A scale factor is a number that tells you how much to enlarge or reduce a shape in a dilation. Here are a couple of examples to illustrate scale factors:

### Example 1: Enlargement
- **Original Segment Length**: Suppose you have a line segment \( \overline{AB} \) with a length of \( 4 \) units.
- **Dilation Resulting Length**: After dilation with a center of dilation at point \( B \), the new segment \( \overline{AC} \) has a length of \( 8 \) units.
- **Scale Factor Calculation**:
\[
k = \frac{\text{Length of } \overline{AC}}{\text{Length of } \overline{AB}} = \frac{8}{4} = 2
\]
In this example, the scale factor is \( 2 \), meaning the segment was enlarged to twice its original length.

### Example 2: Reduction
- **Original Segment Length**: Suppose you have a line segment \( \overline{AB} \) with a length of \( 6 \) units.
- **Dilation Resulting Length**: After dilation, the new segment \( \overline{AC} \) has a length of \( 3 \) units.
- **Scale Factor Calculation**:
\[
k = \frac{\text{Length of } \overline{AC}}{\text{Length of } \overline{AB}} = \frac{3}{6} = \frac{1}{2}
\]
In this example, the scale factor is \( \frac{1}{2} \), meaning the segment was reduced to half its original length.

### Summary
- A scale factor greater than \( 1 \) indicates an enlargement.
- A scale factor less than \( 1 \) indicates a reduction.
- A scale factor of \( 1 \) means there is no change in size.

Use these examples to determine the scale factor in your own problem! If you have specific values or points from your problem, I can help you calculate the exact scale factor.