Triangle PQR Has Sides Measuring 9 Feet And 10 Feet, And A Perimeter Of 24 Feet. What Is The Area Of Triangle PQR? Round To The Nearest Square Foot.Use Heron's Formula: Area = S ( S − A ) ( S − B ) ( S − C ) \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} Area = S ( S − A ) ( S − B ) ( S − C ) Possible Answers:A. 6 Square FeetB.

Mar 11, 2025 by ADMIN 323 views

**Solving for the Area of Triangle PQR Using Heron's Formula**

Introduction to Heron's Formula

Heron's formula is a mathematical formula used to calculate the area of a triangle when the lengths of all three sides are known. The formula is given by $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ , where $a$ , $b$ , and $c$ are the lengths of the sides of the triangle, and $s$ is the semi-perimeter, which is half the perimeter of the triangle. In this article, we will use Heron's formula to find the area of triangle PQR, given that its sides measure 9 feet and 10 feet, and its perimeter is 24 feet.

Calculating the Semi-Perimeter of Triangle PQR

To use Heron's formula, we first need to calculate the semi-perimeter of the triangle. The perimeter of the triangle is given as 24 feet, so we can calculate the semi-perimeter as follows:

$s = \frac{P}{2} = \frac{24}{2} = 12$

Applying Heron's Formula to Find the Area of Triangle PQR

Now that we have the semi-perimeter, we can apply Heron's formula to find the area of the triangle. We will substitute the values of $a$ , $b$ , $c$ , and $s$ into the formula and calculate the area:

$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ $= \sqrt{12(12-9)(12-10)(12-6)}$ $= \sqrt{12(3)(2)(6)}$ $= \sqrt{432}$

Rounding the Area to the Nearest Square Foot

The area of the triangle is $\sqrt{432}$ square feet. To round this value to the nearest square foot, we can calculate the square root of 432 and round it to the nearest whole number:

$\sqrt{432} \approx 20.78$

Rounding this value to the nearest square foot, we get:

$\text{Area} \approx 21$ square feet

Conclusion

In this article, we used Heron's formula to find the area of triangle PQR, given that its sides measure 9 feet and 10 feet, and its perimeter is 24 feet. We calculated the semi-perimeter of the triangle and applied Heron's formula to find the area. Finally, we rounded the area to the nearest square foot to get a final answer of 21 square feet.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Calculate the semi-perimeter of the triangle: $s = \frac{P}{2} = \frac{24}{2} = 12$
Apply Heron's formula to find the area of the triangle: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ $= \sqrt{12(12-9)(12-10)(12-6)}$ $= \sqrt{12(3)(2)(6)}$ $= \sqrt{432}$
Round the area to the nearest square foot: $\sqrt{432} \approx 20.78$ $\text{Area} \approx 21$ square feet

Possible Answers

A. 6 square feet B. 21 square feet

The correct answer is B. 21 square feet.

Introduction to Heron's Formula FAQ

Heron's formula is a mathematical formula used to calculate the area of a triangle when the lengths of all three sides are known. In this article, we will answer some frequently asked questions about Heron's formula, including how to use it, what it is used for, and how to calculate the semi-perimeter of a triangle.

Q: What is Heron's Formula?

A: Heron's formula is a mathematical formula used to calculate the area of a triangle when the lengths of all three sides are known. The formula is given by $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ , where $a$ , $b$ , and $c$ are the lengths of the sides of the triangle, and $s$ is the semi-perimeter, which is half the perimeter of the triangle.

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 by dividing the perimeter of the triangle by 2. For example, if the perimeter of a triangle is 24 feet, the semi-perimeter would be 12 feet.

Q: How Do I Use Heron's Formula to Calculate the Area of a Triangle?

A: To use Heron's formula, you need to know the lengths of all three sides of the triangle and the perimeter of the triangle. You can then calculate the semi-perimeter by dividing the perimeter by 2. Next, you can substitute the values of $a$ , $b$ , $c$ , and $s$ into the formula and calculate the area.

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

A: The formula for the semi-perimeter of a triangle is $s = \frac{P}{2}$ , where $P$ is the perimeter of the triangle.

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

A: Yes, you can use Heron's formula to calculate 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 for the area of a right triangle, which is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .

Q: What is the Difference Between Heron's Formula and the Formula for the Area of a Right Triangle?

A: Heron's formula is used to calculate the area of any triangle, regardless of whether it is a right triangle or not. The formula for the area of a right triangle is used specifically for right triangles and is given by $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .

Q: Can I Use Heron's Formula to Calculate the Area of an Isosceles Triangle?

A: Yes, you can use Heron's formula to calculate the area of an isosceles triangle. However, if you know the lengths of the two equal sides of the isosceles triangle and the base, you can use the formula for the area of an isosceles triangle, which is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .

Q: What is the Formula for the Area of an Isosceles Triangle?

A: The formula for the area of an isosceles triangle is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .