To determine whether the dashed triangle is an enlargement or a reduction compared to the solid triangle and to find the scale factor, we can use the following steps:
1. **Identify Points**: Identify corresponding points on the solid and dashed triangles.
2. **Calculate Distances**: Measure the distances from the origin (the center of dilation) to these corresponding points on both triangles.
3. **Compare Distances**: Compare the distances of the corresponding points to understand how the size of the dashed triangle relates to the solid triangle.
Let's illustrate this with a hypothetical example (since we don’t have the actual image):
Suppose you identify a point on the solid triangle at coordinates \( (x_1, y_1) \) and the corresponding point on the dashed triangle at \( (x_2, y_2) \).
Let’s say the distance from the origin to the point on the solid triangle (\( (x_1, y_1) \)) is \( d_1 \):
\[ d_1 = \sqrt{x_1^2 + y_1^2} \]
And the distance from the origin to the corresponding point on the dashed triangle (\( (x_2, y_2) \)) is \( d_2 \):
\[ d_2 = \sqrt{x_2^2 + y_2^2} \]
4. **Determine the Scale Factor**:
\[ \text{Scale Factor} = \frac{d_2}{d_1} \]
**Enlargement or Reduction**:
- If the scale factor is greater than 1 (\( d_2 > d_1 \)), the dilation is an enlargement.
- If the scale factor is less than 1 (\( d_2 < d_1 \)), the dilation is a reduction.
Given the options provided:
- Enlargement, Scale factor of 3
- Enlargement, Scale factor of 1/3
- Reduction, scale factor of 3
- Reduction, scale factor of 1/3
**Example Calculation**:
If \( d_2 = 3 \) and \( d_1 = 1 \):
\[ \text{Scale Factor} = \frac{3}{1} = 3 \]
This means the dashed triangle is an enlargement with a scale factor of 3.
If \( d_2 = 1 \) and \( d_1 = 3 \):
\[ \text{Scale Factor} = \frac{1}{3} \]
This means the dashed triangle is a reduction with a scale factor of \( \frac{1}{3} \).
To select the correct answer:
- Based on whether \( d_2 \) is smaller or larger than \( d_1 \), you choose either "Reduction" or "Enlargement".
- Use the calculated scale factor to choose whether it’s 3 or 1/3.
Therefore, if the areas or distances indeed show the dashed triangle is smaller compared to the solid triangle, then the correct choice is:
- **Reduction, scale factor of 1/3**
13) The dashed triangle is a dilation image of the solid triangle with the center at the origin. Is the dilation an enlargement or a reduction? Find the scale factor of the dilation. (Select only one answer)
Captionless Image
Enlargement, Scale factor of 3
Enlargement, Scale factor of 1/3
Reduction, scale factor of 3
Reduction, scale factor of 1/3
1 answer