Problem 30 - Precalculus Mathematics for Calculus

Question

Find the area of the triangle whose sides have the given lengths. \(a=1, \quad b=2, \quad c=2\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 0.968 square units.

Step 1: Verify the Triangle Inequality Theorem

Before calculating the area, verify that a triangle can exist with the given side lengths. The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. Check the following:1. \(a + b > c \rightarrow 1 + 2 > 2\) ✔️2. \(a + c > b \rightarrow 1 + 2 > 2\) ✔️3. \(b + c > a \rightarrow 2 + 2 > 1\) ✔️Since all conditions are satisfied, we can proceed to find the area.

Step 2: Calculate the Semi-Perimeter

Calculate the semi-perimeter \(s\) of the triangle using the formula: \[ s = \frac{a + b + c}{2} \]Substitute the given values: \[ s = \frac{1 + 2 + 2}{2} = \frac{5}{2} = 2.5 \]

Step 3: Apply Heron's Formula

Use Heron's formula to find the area \(A\) of the triangle. Heron's formula is:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute \(s = 2.5\), \(a = 1\), \(b = 2\), \(c = 2\):\[ A = \sqrt{2.5(2.5-1)(2.5-2)(2.5-2)} \]

Step 4: Simplify Heron's Formula Expression

Continue simplifying the expression for the area:\[ A = \sqrt{2.5 \times 1.5 \times 0.5 \times 0.5} \]\[ A = \sqrt{2.5 \times 0.375} \]\[ A = \sqrt{0.9375} \]

Step 5: Calculate the Area Value

Calculate the square root of the simplified expression:\[ A \approx \sqrt{0.9375} = 0.9682458 \]Thus, the area of the triangle is approximately \(0.968\) square units.

Key Concepts

Heron's Formula

Heron's formula is a useful tool for finding the area of a triangle when you know the lengths of all three sides. It's especially helpful for triangles that aren't right-angled. This formula uses the semi-perimeter of the triangle to calculate the area.
Heron's formula is given by:

• \( A = \sqrt{s(s-a)(s-b)(s-c)} \)
• Where \( A \) is the area of the triangle.
• \( s \) is the semi-perimeter of the triangle.
• \( a \), \( b \), and \( c \) are the lengths of the triangle's sides.

To apply Heron's formula:

• First, calculate the semi-perimeter using the formula \( s = \frac{a + b + c}{2} \).
• Next, substitute the known side lengths and the semi-perimeter into the formula to find the area.

This is a powerful method because it allows calculating area from just side lengths. No need for angles or heights, making it versatile for various triangle types.

Triangle Inequality Theorem

The Triangle Inequality Theorem is essential when confirming whether three given lengths can form a triangle. It's a necessary step before calculating a triangle's area to ensure the side lengths are feasible.
The theorem states:

• The sum of the lengths of any two sides must be greater than the length of the third side.

For three sides \(a, b, \text{ and } c\):

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

If all these conditions hold true, the lengths can form a triangle. This theorem ensures the sides meet the structural requirements of a triangle, preventing invalid or impossible configurations. Once confirmed, we can proceed with calculations like using Heron's formula for area.

Semi-Perimeter Calculation

The semi-perimeter is a crucial intermediary in calculating a triangle's area using Heron's formula. It simplifies the process and serves as a stepping stone for the formula itself.
The semi-perimeter \(s\) is calculated as follows:

• \( s = \frac{a + b + c}{2} \)
• Where: \(a, b, \text{ and } c\) are the side lengths of the triangle.

By averaging the perimeter of the triangle, we gain a consistent value that is easy to work with in further calculations. This concept is not just practical in Heron's formula; it also helps build intuition around the geometry of triangles. It's accessible to compute and serves as a basis for more complex geometric analysis. Hence, always calculate the semi-perimeter when dealing with side-based problems in triangles.