Question
What do you know about the scale factor for the dilation shown? Explain.
Triangle A B C is reduced to triangle A prime B prime C prime.
Triangle A B C is reduced to triangle A prime B prime C prime.
Answers
GPT-4o mini
In the context of geometry, a dilation is a transformation that produces a figure similar to the original figure but with a different size. The scale factor for a dilation is the ratio of a side length of the image (the dilated figure) to the corresponding side length of the original figure.
In your case, you mentioned that triangle ABC has been reduced to triangle A'B'C'. This indicates that the scale factor is less than 1 (i.e., the second triangle is smaller than the first).
To find the scale factor for the dilation from triangle ABC to triangle A'B'C', you can use the following steps:
1. **Identify corresponding side lengths**: Choose a side length from triangle ABC (for example, side AB) and find its corresponding side length in triangle A'B'C' (AB' in this case).
2. **Calculate the scale factor**: Divide the length of the side in the image by the length of the corresponding side in the original triangle.
\[
\text{Scale Factor} = \frac{\text{Length of side in A'B'C'}}{\text{Length of side in ABC}}
\]
3. **Interpret the scale factor**: If the scale factor is between 0 and 1, it confirms that triangle A'B'C' is a reduced version of triangle ABC. If the scale factor is greater than 1, it would imply an enlargement, which is not the case here.
In summary, for triangle ABC being reduced to triangle A'B'C', the scale factor will be a positive fraction less than 1, indicating that all dimensions of the triangle have been reduced by that factor.
In your case, you mentioned that triangle ABC has been reduced to triangle A'B'C'. This indicates that the scale factor is less than 1 (i.e., the second triangle is smaller than the first).
To find the scale factor for the dilation from triangle ABC to triangle A'B'C', you can use the following steps:
1. **Identify corresponding side lengths**: Choose a side length from triangle ABC (for example, side AB) and find its corresponding side length in triangle A'B'C' (AB' in this case).
2. **Calculate the scale factor**: Divide the length of the side in the image by the length of the corresponding side in the original triangle.
\[
\text{Scale Factor} = \frac{\text{Length of side in A'B'C'}}{\text{Length of side in ABC}}
\]
3. **Interpret the scale factor**: If the scale factor is between 0 and 1, it confirms that triangle A'B'C' is a reduced version of triangle ABC. If the scale factor is greater than 1, it would imply an enlargement, which is not the case here.
In summary, for triangle ABC being reduced to triangle A'B'C', the scale factor will be a positive fraction less than 1, indicating that all dimensions of the triangle have been reduced by that factor.