https://file.aiquickdraw.com/mathbot/file/1729277391919ysf7k5sc.jpg

The given formula represents Heron's formula for the area \( A \) of a triangle when the lengths of its three sides \( a \), \( b \), and \( c \) are known, along with the semi-perimeter \( s \):

\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]

The problem provides the following values:
- \( a = 15 \frac{1}{2} \) inches (which is \( a = 15.5 \) inches),
- \( b = 13 \frac{1}{2} \) inches (which is \( b = 13.5 \) inches),
- \( c = 12 \frac{1}{4} \) inches (which is \( c = 12.25 \) inches),
- \( s = 20 \) inches.

### Step-by-step solution:

1. **Subtract each side from the semi-perimeter**:
   - \( s - a = 20 - 15.5 = 4.5 \)
   - \( s - b = 20 - 13.5 = 6.5 \)
   - \( s - c = 20 - 12.25 = 7.75 \)

2. **Substitute all the values into the formula**:
   \[
   A = \sqrt{20(4.5)(6.5)(7.75)}
   \]

3. **Calculate the product**:
   \[
   20 \times 4.5 = 90
   \]
   \[
   90 \times 6.5 = 585
   \]
   \[
   585 \times 7.75 = 4533.75
   \]

4. **Take the square root**:
   \[
   A = \sqrt{4533.75} \approx 67.32 \, \text{square inches}
   \]

Thus, the area of the triangle is approximately \( A \approx 67.32 \, \text{square inches} \).

Would you like more details on any of the steps, or further clarifications?

Here are 5 related questions for further exploration:
1. How would the area change if the semi-perimeter were different?
2. What does the semi-perimeter represent geometrically?
3. Can Heron’s formula be used for a right triangle, and how does it compare with the usual area formula for such triangles?
4. What would the area be if the triangle were equilateral, with all sides equal?
5. How do we handle a situation where the semi-perimeter is smaller than one of the sides?

**Tip:** When using Heron's formula, always ensure that the sum of any two sides is greater than the third, as required by the triangle inequality.

Evaluate the given formula A = sqrt(s(s-a)(s-b)(s-c)) for a = 15 1/2 in., b = 13 1/2 in., c = 12 1/4 in., s = 20 in.

Learn how to calculate the area of a triangle using Heron's formula with side lengths of a = 15.5 in, b = 13.5 in, and c = 12.25 in. The solution involves step-by-step calculations to find the area, approximately 67.32 square inches. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation