Question
Unit Posttest Select the correct answer. Pentagon \( A B C D E \) is similar to pentagon \( P Q R S T \). If the side length of pentagon \( A B C D E \) is 6 times the side length of pentagon \( P Q R S T \), which statement is true? A. The area of pentagon \( A B C D E \) is 6 times the area of pentagon \( P Q R S T \). The area of pentagon \( A B C D E \) is 12 times the area of pentagon \( P Q R S T \). C. The area of pentagon \( A B C D E \) is 36 times the area of pentagon \( P Q R S T \). D. The area of pentagon \( A B C D E \) is 216 times the area of pentagon \( P Q R S T \).
Ask by Peterson Wyatt. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The area of pentagon \( A B C D E \) is 36 times the area of pentagon \( P Q R S T \).
Solution
To determine the correct statement about the areas of the similar pentagons, let's analyze the given information:
- **Pentagons \( A B C D E \) and \( P Q R S T \) are similar.**
- **The side length of pentagon \( A B C D E \) is 6 times the side length of pentagon \( P Q R S T \).**
### Understanding Similarity and Area Scaling
When two polygons are similar, the ratio of their areas is the square of the ratio of their corresponding side lengths. This relationship can be expressed as:
\[
\text{Area Ratio} = \left(\frac{\text{Side Length of } A}{\text{Side Length of } P}\right)^2
\]
Given that the side length of \( A B C D E \) is 6 times that of \( P Q R S T \):
\[
\text{Area Ratio} = 6^2 = 36
\]
### Conclusion
The area of pentagon \( A B C D E \) is **36 times** the area of pentagon \( P Q R S T \).
**Correct Answer:**
C. The area of pentagon \( A B C D E \) is 36 times the area of pentagon \( P Q R S T \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
When two figures are similar, the ratio of their corresponding side lengths is also the ratio of the square roots of their areas. In this case, since the side length of pentagon \( A B C D E \) is 6 times that of pentagon \( P Q R S T \), you would square that ratio to find the relationship between their areas. Therefore, if the sides are in the ratio of 6:1, the areas will be in the ratio of \( 6^2:1^2 \), which is \( 36:1 \). Thus, the correct statement is C. The area of pentagon \( A B C D E \) is 36 times the area of pentagon \( P Q R S T \).
