Problem 27 - Algebra and Trigonometry

Question

Find the area of the triangle whose sides have the given lengths. \(a=9, \quad b=12, \quad c=15\)

Step-by-Step Solution

Answer

The area of the triangle is 54 square units.

Step 1: Determine Semi-Perimeter

First, calculate the semi-perimeter of the triangle using the formula: \( s = \frac{a+b+c}{2} \). Substitute the values: \( s = \frac{9+12+15}{2} = 18 \).

Step 2: Apply Heron's Formula

Use Heron's formula for the area of a triangle: \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Substitute the values for \( s \), \( a \), \( b \), and \( c \): \( A = \sqrt{18(18-9)(18-12)(18-15)} \).

Step 3: Simplify Inside the Square Root

Calculate inside the square root: \( A = \sqrt{18 \times 9 \times 6 \times 3} \).

Step 4: Calculate Area

Simplify and calculate the expression: \( 18 \times 9 \times 6 \times 3 = 2916 \). Then, \( A = \sqrt{2916} = 54 \).

Key Concepts

Semi-Perimeter

The semi-perimeter is an important concept when calculating the area of a triangle using Heron's Formula. It is half of the triangle's perimeter. Here's why it matters and how to calculate it easily.
To find the semi-perimeter, use the formula:

• First, sum the lengths of all three sides: \( a + b + c \). In our example, the sides have the lengths 9, 12, and 15.
• Calculate the total: \( 9 + 12 + 15 = 36 \).
• Divide this sum by 2 to get the semi-perimeter: \( \frac{36}{2} = 18 \).

The semi-perimeter is a crucial step in efficiently using Heron's Formula. Remember, the value of the semi-perimeter helps you simplify the process of finding the area of the triangle.

Area of Triangle

Heron's Formula is a fantastic method for finding the area of a triangle when you know the lengths of all three sides. You don't need the height, making this approach versatile and easy.
Apply Heron’s Formula using:

• First, calculate the semi-perimeter, \( s \).
• Next, calculate \( s-a \), \( s-b \), and \( s-c \):
  • \( s-a = 18 - 9 = 9 \)
  • \( s-b = 18 - 12 = 6 \)
  • \( s-c = 18 - 15 = 3 \)
• Plug these values into the formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
• Use the values in the example: \[ A = \sqrt{18 \times 9 \times 6 \times 3} \]

This calculation provides the complete picture of how Heron’s Formula works, giving you the triangle's area swiftly and effectively. In our example, the computed area is 54 square units.

Square Root Calculation

Calculating the square root is a fundamental skill in mathematics, especially when using Heron’s Formula to find the area of a triangle. Performing the calculation accurately ensures you get the correct area.
Here's a simple guide on calculating square roots, particularly in the context of Heron’s Formula:

• First, calculate the expression under the square root: \( 18 \times 9 \times 6 \times 3 = 2916 \).
• Now, compute the square root of this expression: \( \sqrt{2916} \).
• The calculated square root, in this example, results in 54.

Mastering the square root calculation is essential to determine the area of a triangle successfully, as it finalizes the process. In mathematical problems, ensuring each step is performed correctly helps achieve accurate outcomes.