Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Heron's Area Formula to find the area of the triangle. $$a=1, \quad b=\frac{1}{2}, \quad c=\frac{3}{4}$$

Step-by-Step Solution

Answer

The area of the triangle, calculated using Heron's formula, is approximately 0.055.

Step 1: Calculate the Semi-perimeter

First, add all three given side lengths and divide by 2. This is called the semi-perimeter (s) of the triangle. Using the given lengths, \(a = 1\), \(b = \frac{1}{2}\), and \(c = \frac{3}{4}\), you get \(s = \frac{1+ \frac{1}{2} + \frac{3}{4}}{2} = \frac{1.125}{2} = 0.5625\).

Step 2: Apply Heron's Formula

The Heron's formula for the area of a triangle with side lengths \(a\), \(b\), and \(c\) and semi-perimeter \(s\) is: \[Area = \sqrt{s*(s-a)*(s-b)*(s-c)}\] Now insert the calculated semi-perimeter and the given side lengths into the equation: \[Area = \sqrt{0.5625*(0.5625 - 1)*(0.5625 - \frac{1}{2})*(0.5625-\frac{3}{4})}\]

Step 3: Solve The Equation

Solve the expression within the square root first, followed by calculating the square root itself, this will provide you with the area of the triangle: \[Area = \sqrt{0.5625*(-0.4375)*0.0625*(-0.1875)} = \sqrt{0.0030517578125} = 0.0552259\]