Problem 45 - Trigonometry

Question

If Heron's formula is used to find the area of triangle \(A B C\) having \(a=3\) meters, \(b=5\) meters, and \(c=6\) meters, which of the following shows the correct way to set up the formula? a. \(S=7 \sqrt{(10)(12)(13)}\) b. \(S=\sqrt{(4)(2)(1)}\) c. \(S=\sqrt{7(3)(5)(6)}\) d. \(S=\sqrt{7(4)(2)(1)}\)

Step-by-Step Solution

Answer

The correct setup according to Heron's formula is option (d): \( S = \sqrt{7(4)(2)(1)} \).

Step 1: Recall Heron's Formula

Heron's formula is used to find the area of a triangle when the lengths of all three sides are known. The formula is: \( S = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s \) is the semi-perimeter of the triangle, calculated as \( s = \frac{a+b+c}{2} \).

Step 2: Calculate the Semi-Perimeter

First, find the semi-perimeter \( s \) of triangle \( ABC \) with sides \( a = 3 \), \( b = 5 \), and \( c = 6 \). \[ s = \frac{3 + 5 + 6}{2} = \frac{14}{2} = 7 \].

Step 3: Apply Heron's Formula

Now that we have \( s = 7 \), substitute \( s \), \( a \), \( b \), and \( c \) into the Heron's formula: \[ S = \sqrt{7(7-3)(7-5)(7-6)} = \sqrt{7 \times 4 \times 2 \times 1} \].

Step 4: Compare the Options

Compare the result from Step 3 to the given options. Option (d) matches the setup we found: \( S = \sqrt{7(4)(2)(1)} \).

Key Concepts

Semi-Perimeter of a Triangle

The semi-perimeter of a triangle is a crucial concept used to simplify problems involving both perimeter and area. When you're given the sides of a triangle, like in our exercise with sides \( a = 3 \) m, \( b = 5 \) m, and \( c = 6 \) m, you first calculate the perimeter by adding up all the side lengths. The semi-perimeter, denoted as \( s \), is simply half of this total perimeter.
For triangle \( ABC \), the total perimeter is \( 3 + 5 + 6 = 14 \) meters. So, the semi-perimeter \( s \) is \( \frac{14}{2} = 7 \) meters.
Why is this important? Using the semi-perimeter reduces the complexity when using formulas like Heron's formula. This value of \( s \) becomes a starting point to further explore the properties of a triangle, like its area using Heron's formula.
Understanding the semi-perimeter helps build an intuitive grasp of a triangle's dimension and its potential transformations in applied problems.

Area of a Triangle

Finding the area of a triangle is a fundamental challenge in geometry, especially when you only have the side lengths. Fortunately, Heron's formula provides a way to calculate the area without needing to know heights or angles.
Heron's formula is expressed as:

• \( S = \sqrt{s(s-a)(s-b)(s-c)} \)

This formula uses the semi-perimeter \( s \), as well as the side lengths \( a \), \( b \), and \( c \).
In our example with triangle \( ABC \), we have already calculated the semi-perimeter \( s = 7 \). Next, we substitute into Heron's formula: \[ S = \sqrt{7(7-3)(7-5)(7-6)} \] This simplifies to: \[ S = \sqrt{7 \times 4 \times 2 \times 1} \] Using Heron's formula is a powerful method because it allows you to find the area regardless of additional information like height. All you need are the side lengths and the semi-perimeter.

Triangle Side Lengths

Understanding the role of triangle side lengths is foundational for solving a variety of problems in geometry. Each of the three sides provides a piece of the puzzle required to unlock properties like area, perimeter, and more.
In the given triangle \( ABC \) with sides \( a = 3 \) m, \( b = 5 \) m, and \( c = 6 \) m, these lengths not only tell us about the boundaries of the triangle but also help us reach other important geometric conclusions.
When calculating the semi-perimeter (\( s \)) and using Heron's formula, every side length is essential. They determine the semi-perimeter \( s \) as \( \frac{a+b+c}{2} \), which serves as a stepping-stone to further calculations.
The side lengths become input for Heron's formula, with each subtraction \((s-a), (s-b), (s-c)\) forming part of the calculation for the area. Knowing each side length allows us to explore not just the area via Heron's formula, but offers insight into the triangle's type, potential symmetry, and its physical representation.