https://file.aiquickdraw.com/mathbot/file/1727252561346bsyw8frn.jpg

To solve this problem, we can use the fact that pentagons \( A \) and \( B \) are similar. For similar shapes, the ratio of corresponding lengths is the square root of the ratio of their areas.

### Step 1: Find the ratio of the areas
We are given:
- The area of pentagon \( A \) is \( 96 \, \text{cm}^2 \),
- The area of pentagon \( B \) is \( 253.5 \, \text{cm}^2 \).

The ratio of the areas is:
\[
\frac{\text{Area of } B}{\text{Area of } A} = \frac{253.5}{96}
\]

### Step 2: Find the square root of the area ratio
Since the ratio of corresponding lengths is the square root of the area ratio, we need to compute:
\[
\text{Ratio of lengths} = \sqrt{\frac{253.5}{96}}
\]

### Step 3: Use the ratio to find \( x \)
We are given that the base of pentagon \( B \) is 13 cm, and we need to find the corresponding base \( x \) of pentagon \( A \). Using the length ratio, we set up the proportion:
\[
\frac{x}{13} = \sqrt{\frac{96}{253.5}}
\]
Solving this will give us the value of \( x \).

Let me calculate this for you.The length marked \( x \) in the diagram is \( 8 \, \text{cm} \).

Would you like further details on this solution or have any questions?

Here are 5 related questions to expand on this:

1. How do you find the ratio of areas for other types of polygons?
2. Why does the square root of the area ratio give the length ratio?
3. What if the shapes were not similar—how would the approach change?
4. How can we generalize this method for 3D shapes?
5. How does the concept of similarity apply to angles in polygons?

**Tip**: Always double-check whether the shapes are similar before using ratios, as this method only applies to similar figures!

Pentagons A and B are similar. The area of pentagon A is 96 cm². The area of pentagon B is 253.5 cm². The base of pentagon B is 13 cm. Find the length marked x in the diagram.

Learn how to solve a problem involving similar pentagons, where you find a missing length using the ratio of areas and proportions. Given the areas of pentagons A (96 cm²) and B (253.5 cm²), this solution explains step-by-step how to find the length of the base of pentagon A when pentagon B's base is known. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation