The parallelogram S'T'U'V' is a dilation of the parallelogram STUV. What is the scale factor of the dilation?
Understand the Problem
The question asks for the scale factor of a dilation given two parallelograms, S'T'U'V' and STUV, where S'T'U'V' is the dilated image of STUV. We need to determine the ratio of corresponding side lengths to find the scale factor.
Answer
$\frac{1}{3}$
Answer for screen readers
The scale factor of the dilation is $\frac{1}{3}$.
Steps to Solve
-
Identify corresponding sides
We need to find two corresponding sides in the original parallelogram $STUV$ and the dilated parallelogram $S'T'U'V'$. We are given that $S'T'U'V'$ is the image of $STUV$. Therefore, side $ST$ corresponds to side $S'T'$. We have $ST = 9$ and $S'T' = 3$.
-
Calculate the scale factor
The scale factor of a dilation is the ratio of the length of a side in the image to the length of the corresponding side in the original figure. In this case, the scale factor $k$ is given by:
$$ k = \frac{S'T'}{ST} $$
Substituting the given values:
$$ k = \frac{3}{9} $$
-
Simplify the scale factor
Simplify the fraction to find the scale factor:
$$ k = \frac{1}{3} $$
The scale factor of the dilation is $\frac{1}{3}$.
More Information
A scale factor of $\frac{1}{3}$ means that the image $S'T'U'V'$ is smaller than the original parallelogram $STUV$. This is a reduction or contraction.
Tips
A common mistake is to divide the length of the side in the original figure by the length of the corresponding side in the image, resulting in the reciprocal of the actual scale factor. Remember that the scale factor is always $\frac{\text{image}}{\text{original}}$.
AI-generated content may contain errors. Please verify critical information
