Problem 31 - Precalculus Mathematics for Calculus

Question

Find the area of the triangle whose sides have the given lengths. \(a=7, \quad b=8, \quad c=9\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 26.83 square units.

Step 1: Calculate the Semi-Perimeter

First, calculate the semi-perimeter of the triangle, which is half the sum of all sides. The formula for the semi-perimeter \( s \) is: \[ s = \frac{a + b + c}{2} \]Substitute the values of \( a, b, \) and \( c \) into the formula:\[ s = \frac{7 + 8 + 9}{2} = 12 \]

Step 2: Apply Heron's Formula

Now apply Heron’s Formula to calculate the area of the triangle. Heron's Formula is given by:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute our values:\( s = 12, a = 7, b = 8, \) and \( c = 9 \).

Step 3: Substitute Values into Heron's Formula

Substitute the semi-perimeter and the side lengths into Heron's Formula:\[A = \sqrt{12(12-7)(12-8)(12-9)}\]This simplifies to:\[A = \sqrt{12 \times 5 \times 4 \times 3}\]

Step 4: Calculate Inside the Square Root

Carry out the multiplication inside the square root:\[A = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720}\]

Step 5: Simplify the Square Root

Calculate the square root:\[A = \sqrt{720} \approx 26.83\]Thus, the area of the triangle is approximately 26.83 square units.

Key Concepts

Triangle Area

Calculating the area of a triangle is a fundamental concept in geometry. One effective method for triangles with known side lengths is Heron's Formula. Unlike the traditional base-height formula, which requires a perpendicular height, Heron's Formula relies solely on the lengths of the triangle's sides. This makes it particularly useful for irregular triangles, where determining the height might be challenging. By using Heron's Formula, you can conveniently find the area without additional geometric constraints. Once you have the semi-perimeter, you can use the lengths of the sides directly. You plug these values into Heron's Formula to get the area. It's a valuable tool that simplifies working with triangles. Using this formula, we can understand how all elements of a triangle interact, offering insights into spatial relationships and properties.

Semi-Perimeter

Before we can use Heron's Formula, we need to calculate the semi-perimeter of the triangle. This is the first crucial step in the process. The semi-perimeter is simply half the perimeter of the triangle. The formula is straightforward: add up all the sides and then divide by two. Mathematically, it is expressed as: \[ s = \frac{a + b + c}{2} \] where \( a, b, \) and \( c \) are the lengths of the sides of the triangle. Calculating the semi-perimeter simplifies the task of evaluating Heron's Formula later. For our example, with sides \( a = 7, b = 8, \) and \( c = 9 \), the semi-perimeter \( s \) is \( 12 \). Understanding and mastering this concept is crucial because it provides the foundation for calculating the area.

Square Root Calculation

The final step in using Heron's Formula involves calculating the square root. After determining the product \( s(s-a)(s-b)(s-c) \), the last task is to find the square root of this result. For our triangle, the expression inside the square root simplifies to \( 720 \). Calculating the square root of a number can often be done using a calculator for precision. In this case, \( \sqrt{720} \approx 26.83 \). It's important to simplify inside the square root first to avoid complex calculations. Understanding how to calculate the square root and simplifying expressions is key to obtaining the correct area. Square roots can seem daunting, but they are just another tool to solve the puzzle that is geometry. This method allows us to see the hidden relationships between number operations and the geometry of shapes.