Determining the Length of the Third Side

Oct 13, 2025 by wbrum.org 108 views

Introduction

In geometry, Heron's formula is a widely used method for calculating the area of a triangle when all three sides are known. This formula is particularly useful in situations where the height of the triangle is not easily determined. In this article, we will apply Heron's formula to find the area of triangle PQR, given the lengths of two sides and the unknown length of the third side.

Understanding Heron's Formula

Heron's formula is given by the equation:

Area $= \sqrt{s(s-a)(s-b)(s-c)}$

where:

$a$ , $b$ , and $c$ are the lengths of the three sides of the triangle
$s$ is the semi-perimeter of the triangle, calculated as $s = \frac{a + b + c}{2}$

Calculating the Semi-Perimeter

To apply Heron's formula, we first need to calculate the semi-perimeter of the triangle. We are given the lengths of two sides: $a = 9$ feet and $b = 10$ feet. Let's assume the length of the third side is $c$ . The semi-perimeter is given by:

$s = \frac{a + b + c}{2}$

We will use this equation to calculate the semi-perimeter once we have determined the value of $c$ .

Determining the Length of the Third Side

To find the length of the third side, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have:

$a + b > c$ $9 + 10 > c$ $c < 19$

Similarly, we can write:

$a + c > b$ $9 + c > 10$ $c > 1$

We can also write:

$b + c > a$ $10 + c > 9$ $c > -1$

Since $c$ must be a positive length, we can conclude that:

$1 < c < 19$

Finding the Area Using Heron's Formula

Now that we have determined the range of possible values for $c$ , we can use Heron's formula to find the area of the triangle. We will assume that the length of the third side is $c = 9$ feet, which is the smallest possible value within the range.

First, we calculate the semi-perimeter:

$s = \frac{a + b + c}{2}$ $s = \frac{9 + 10 + 9}{2}$ $s = 14$

Next, we plug the values into Heron's formula:

Area $= \sqrt{s(s-a)(s-b)(s-c)}$ Area $= \sqrt{14(14-9)(14-10)(14-9)}$ Area $= \sqrt{14 \times 5 \times 4 \times 5}$ Area $= \sqrt{1400}$ Area $= 37.42$

However, we are asked to round the area to the nearest square foot. Therefore, the area of the triangle is approximately:

19 square feet

Conclusion

In this article, we applied Heron's formula to find the area of triangle PQR, given the lengths of two sides and the unknown length of the third side. We determined the range of possible values for the length of the third side and assumed a value of $c = 9$ feet. Using Heron's formula, we calculated the semi-perimeter and plugged the values into the formula to find the area of the triangle. The area of the triangle is approximately 19 square feet.

Discussion

The triangle inequality theorem is a fundamental concept in geometry that helps us determine the range of possible values for the length of the third side of a triangle. Heron's formula is a powerful tool for calculating the area of a triangle when all three sides are known. In this article, we demonstrated how to apply Heron's formula to find the area of a triangle and rounded the result to the nearest square foot.

References

Heron's Formula. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Heron's_formula
Triangle Inequality Theorem. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Triangle_inequality

Calculating the Area of a Triangle Using the Formula
Understanding the Triangle Inequality Theorem
Applying Heron's Formula to Find the Area of a Triangle
Triangle PQR: Q&A =====================

Q: What is Heron's Formula?

A: Heron's formula is a mathematical formula used to calculate the area of a triangle when all three sides are known. The formula is given by:

Area $= \sqrt{s(s-a)(s-b)(s-c)}$

where:

$a$ , $b$ , and $c$ are the lengths of the three sides of the triangle
$s$ is the semi-perimeter of the triangle, calculated as $s = \frac{a + b + c}{2}$

Q: What is the Semi-Perimeter of a Triangle?

A: The semi-perimeter of a triangle is half the perimeter of the triangle. It is calculated as:

$s = \frac{a + b + c}{2}$

Q: How Do I Apply Heron's Formula?

A: To apply Heron's formula, you need to follow these steps:

Calculate the semi-perimeter of the triangle using the formula $s = \frac{a + b + c}{2}$ .
Plug the values of $a$ , $b$ , $c$ , and $s$ into the formula: Area $= \sqrt{s(s-a)(s-b)(s-c)}$
Simplify the expression and calculate the area.

Q: What is the Triangle Inequality Theorem?

A: The triangle inequality theorem is a fundamental concept in geometry that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words:

$a + b > c$ $a + c > b$ $b + c > a$

Q: How Do I Determine the Length of the Third Side of a Triangle?

A: To determine the length of the third side of a triangle, you can use the triangle inequality theorem. You need to find the range of possible values for the length of the third side and then use Heron's formula to calculate the area of the triangle.

Q: What is the Formula for the Area of a Triangle?

A: The formula for the area of a triangle is given by:

Area $= \frac{1}{2} \times \text{base} \times \text{height}$

However, if you know the lengths of all three sides, you can use Heron's formula to calculate the area.

Q: Can I Use Heron's Formula to Find the Area of a Right Triangle?

A: Yes, you can use Heron's formula to find the area of a right triangle. However, if you know the lengths of the two legs of the right triangle, you can use the formula:

Area $= \frac{1}{2} \times \text{base} \times \text{height}$

to calculate the area more easily.

Q: What are the Advantages of Using Heron's Formula?

A: The advantages of using Heron's formula include:

It allows you to calculate the area of a triangle when all three sides are known.
It is a powerful tool for solving geometry problems.
It is a fundamental concept in mathematics and is used in many real-world applications.

Q: What are the Disadvantages of Using Heron's Formula?

A: The disadvantages of using Heron's formula include:

It can be complex to apply, especially for beginners.
It requires knowledge of the semi-perimeter and the lengths of all three sides.
It may not be the most efficient method for calculating the area of a triangle.

Conclusion

In this article, we answered some common questions about triangle PQR and Heron's formula. We discussed the formula for the area of a triangle, the semi-perimeter, and the triangle inequality theorem. We also provided examples and explanations to help you understand the concepts better. If you have any more questions, feel free to ask!