Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate the area of triangle \(A B C\). $$a=25.0, \quad b=80.0, \quad c=60.0$$

Step-by-Step Solution

Answer

Using Heron's Formula, the area is approximately 750.0 square units.

Step 1: Use Heron's Formula

The area \( A \) of a triangle with side lengths \( a, b, c \) can be found using Heron's formula. First, calculate the semi-perimeter \( s \) using the formula \( s = \frac{a+b+c}{2} \).

Step 2: Calculate the Semi-Perimeter

Substitute the given side lengths into the formula for the semi-perimeter. So, \( s = \frac{25.0 + 80.0 + 60.0}{2} = 82.5 \).

Step 3: Apply Heron's Formula

The formula for the area is \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Substitute \( s = 82.5 \), \( a = 25.0 \), \( b = 80.0 \) and \( c = 60.0 \) into the formula.

Step 4: Simplify the Expression Inside the Square Root

First calculate each of the terms: \((s-a) = (82.5 - 25) = 57.5\), \((s-b) = (82.5 - 80) = 2.5\), and \((s-c) = (82.5 - 60) = 22.5\).

Step 5: Compute the Area

Substitute back into Heron's formula to find the area:\[ A = \sqrt{82.5 \times 57.5 \times 2.5 \times 22.5} \]Calculate this value to find the approximate area.

Key Concepts

Understanding Semi-Perimeter

The semi-perimeter is a key concept when dealing with triangles and specifically when calculating the area using Heron's formula. It is defined as half the sum of the lengths of the sides of a triangle. Here is how you can derive it:

• Take the sum of the lengths of all three sides of the triangle: \(a + b + c\).
• Divide this sum by 2 to get the semi-perimeter \(s\): \(s = \frac{a + b + c}{2}\).

In our example, the given side lengths are 25.0, 80.0, and 60.0. By plugging these into the formula:\[s = \frac{25.0 + 80.0 + 60.0}{2} = 82.5\]You now have a value for the semi-perimeter, which is essential for finding the area using Heron's formula.

Calculating the Area of a Triangle

Finding the area of a triangle, especially if the side lengths are given, can be done efficiently using Heron's formula. This approach does not require any angles of the triangle, making it unique and handy. First, calculate the semi-perimeter \(s\), as described before:

• Set \(s = 82.5\), \(a = 25.0\), \(b = 80.0\), \(c = 60.0\).
• Heron's formula states the area \(A\) as: \(A = \sqrt{s(s-a)(s-b)(s-c)}\).

Apply these values:

• \((s-a) = (82.5 - 25.0) = 57.5\)
• \((s-b) = (82.5 - 80.0) = 2.5\)
• \((s-c) = (82.5 - 60.0) = 22.5\)

Substitute back into the formula for the final area calculation:\[A = \sqrt{82.5 \times 57.5 \times 2.5 \times 22.5}\]This will give you the approximate area of the triangle.

The Role of Trigonometry Calculations

While Heron's formula itself does not directly involve angles, trigonometry often plays a significant role when dealing with triangles in general. If you had the angles, several trigonometric calculations could assist:

• You could use the sine rule or cosine rule to find missing sides or angles.
• For area, given one angle, you could use the formula: \(A = \frac{1}{2} ab \sin(C)\), where \(C\) is the included angle.

However, when using Heron's formula, the advantage is its reliance purely on side lengths, eliminating the need for angle measurements, hence requiring less trigonometric manipulation. Therefore, while an understanding of trigonometry enhances your grasp of triangle calculations, Heron's formula simplifies the process by focusing on side lengths alone.