Question

Which statements about the dilation are true? Check all that apply.
Triangle X prime Y prime Z prime. Point X prime is 2 units from the center of dilation C and point Z prime is 3 units from the center of dilation. Triangle X Y Z. Point X is 5 units from point C and point Z is 7.5 units from point C.
The center of dilation is point C.
It is a reduction.
It is an enlargement.
The scale factor is 2.5.
The scale factor is Two-fifths.

Answer 1

To analyze the dilation and the statements given, let's break down the information step by step.

1. **Triangle X'Y'Z'**:
- Point X' is 2 units from the center of dilation C.
- Point Z' is 3 units from the center of dilation C.

2. **Triangle XYZ**:
- Point X is 5 units from point C.
- Point Z is 7.5 units from point C.

Now let's determine what type of dilation has occurred and the scale factor.

### Scale Factor Calculation:

The scale factor \( k \) can be found by comparing the distances of the points from the center of dilation.

For point X:
- Original distance: 5 units
- Image distance: 2 units
- Scale factor for X: \( k = \frac{\text{distance from C in triangle XYZ}}{\text{distance from C in triangle X'Y'Z'}} = \frac{5}{2} = 2.5 \)

For point Z:
- Original distance: 7.5 units
- Image distance: 3 units
- Scale factor for Z: \( k = \frac{7.5}{3} = 2.5 \)

### Analysis of Statements:

1. **The center of dilation is point C.**
- **True.** This is given in the problem.

2. **It is a reduction.**
- **False.** A dilation that results in the image being smaller than the original is called a reduction. Since both scale factors calculated (2.5) indicate an enlargement, this statement is false.

3. **It is an enlargement.**
- **True.** Since the scale factors are greater than 1 (2.5), this indicates that the triangles are enlargements.

4. **The scale factor is 2.5.**
- **True.** The calculations show that the scale factor is indeed 2.5.

5. **The scale factor is Two-fifths.**
- **False.** "Two-fifths" represents a scale factor of 0.4, which is less than 1. The actual scale factor is 2.5, not two-fifths.

### Conclusion:

The statements that are true are:
- The center of dilation is point C.
- It is an enlargement.
- The scale factor is 2.5.