Problem 32 - Algebra and Trigonometry

Question

Find the area of the triangle whose sides have the given lengths. \(a=11, \quad b=100, \quad c=101\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 549.36 square units.

Step 1: Calculate the semi-perimeter

The semi-perimeter of a triangle is calculated using the formula \(s = \frac{a + b + c}{2}\). Substituting the given values, we find \(s = \frac{11 + 100 + 101}{2} = 106\).

Step 2: Use Heron's Formula

Heron's Formula is given by \( ext{Area} = \sqrt{s(s-a)(s-b)(s-c)} \). We substitute \(s = 106\), \(a = 11\), \(b = 100\), and \(c = 101\) into the formula: \( ext{Area} = \sqrt{106(106-11)(106-100)(106-101)} \).

Step 3: Simplify Each Term Inside the Square Root

Calculate each term inside the square root. We have: \(106 - 11 = 95\), \(106 - 100 = 6\), \(106 - 101 = 5\). So the formula becomes \( ext{Area} = \sqrt{106 \times 95 \times 6 \times 5} \).

Step 4: Compute the Multiplications

Compute the product \(106 \times 95 \times 6 \times 5\). This product equals \(301,800\).

Step 5: Calculate the Square Root

Take the square root of \(301,800\), which provides the area of the triangle. Therefore, the area is approximately \(549.36\) square units.

Key Concepts

Understanding the Semi-Perimeter

The semi-perimeter plays a fundamental role in calculating the area of a triangle using Heron's Formula. It's defined as half the sum of the lengths of a triangle's sides. This measurement simplifies the process by creating a midpoint based on the perimeter.
The formula for the semi-perimeter is expressed as:

• \( s = \frac{a + b + c}{2} \)

where \( a \), \( b \), and \( c \) represent the lengths of the triangle's sides. By finding the semi-perimeter, you're prepping the elements needed to proceed with Heron's approach.For example, if we have a triangle with sides 11, 100, and 101, the semi-perimeter would be:

• \( s = \frac{11 + 100 + 101}{2} = 106 \)

By setting the stage with this calculation, Heron's Formula becomes much simpler to use in subsequent steps. The semi-perimeter acts as a bridge, linking the perimeter values directly to the area measurement process by being a core part of Heron's Formula.

Triangle Area Calculation Using Heron's Formula

Calculating the area of a triangle using Heron's Formula is an elegant method that leverages the semi-perimeter. Here's the formula you'll be using:

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

When you substitute the values, it fits numbers neatly into a sequence of arithmetic operations under a square root. This strategy is particularly effective when the standard height isn't directly known or isn't easily calculated.Let's walk through a specific example using the sides of a triangle as 11, 100, and 101 with their semi-perimeter, \( s = 106 \):

• Calculate each difference: \( 106 - 11 = 95 \), \( 106 - 100 = 6 \), \( 106 - 101 = 5 \).
• Plug these numbers into Heron's Formula to get: \( \text{Area} = \sqrt{106 \times 95 \times 6 \times 5} \)

This method stands out because it bypasses the traditional base-height area formulation. Instead, it uses only the lengths of the sides, making it very versatile for all types of triangles, whether they're right, obtuse, or acute.

Strengthening Your Mathematical Problem-Solving Skills

Approaching mathematical problems with a clear method like Heron's Formula helps develop strong problem-solving skills. When tackling a problem, it's important to:

• Break down the problem: Start by understanding what's being asked. For area calculations, recognize the type of triangle and use appropriate formulas.
• Apply core formulas: Such as Heron's formula, where understanding the semi-perimeter becomes essential.
• Simplify calculations: Perform arithmetic in steps, calculate differences first before plugging numbers into the formula.
• Check your work: Especially in complex calculations involving square roots or multiplications, revisiting earlier steps ensures accuracy.

In our triangle example, observing the step-by-step setup - from semi-perimeter calculation to executing the formula - exemplifies how structured thinking leads to correct solutions.
Heron’s Formula is not just a tool for solving geometry problems; it is also an exercise in organizing thoughts, executing mathematical operations systematically, and applying formulas effectively. These practices foster a deeper understanding of mathematical concepts and enhance overall problem-solving skills.