How to Calculate the Area of a Triangle with Given Vertices: A Step-by-Step Guide

Calculating the area of a triangle given its vertices is a common problem in geometry. In this article, we will explain the process of finding the area of the triangle ABC with vertices A(1,2), B(-3,1), C(0,1). We will explore different methods including the basic triangle formula, Heron's formula, and the vector and matrix representation method.

Data Points and Calculations

The vertices of triangle ABC are given as A(1,2), B(-3,1), C(0,1). From these points, we can determine the coordinates and lengths of the sides:

BC 3 AB sqrt{17} AC sqrt{2}

Zorthe step-by-step calculations will be provided for each method, including the application of Heron's formula and the matrix determinant approach.

The Basic Triangle Formula

The basic triangle formula for finding the area A of a triangle is given by A (1/2)bh, where b is the base and h is the height of the triangle.

Identify the base b as the distance between points B and C. In this case, b BC 3. Determine the height h as the vertical distance between point A and the line segment BC. Here, h y_A - y_B 2 - 1 1. Substitute b 3 and h 1 into the formula: A (1/2) * 3 * 1 1.5.

This method is straightforward and works well when the base and height are easily identifiable.

Heron's Formula

Heron's formula is a more general method that uses the lengths of the sides of the triangle. If the sides are denoted as a, b, c, and the semi-perimeter is defined as s (a b c) / 2, then the area A is given by:

A sqrt{s(s - a)(s - b)(s - c)}

First, determine the lengths of the sides: a BC 3 b CA sqrt{2} c AB sqrt{17} Calculate the semi-perimeter: s (3 sqrt{2} sqrt{17}) / 2 4.26861 Substitute into Heron's formula: A sqrt{4.26861(4.26861 - 3)(4.26861 - sqrt{2})(4.26861 - sqrt{17})} 1.5

Vectors and Matrices

The matrix and vector method involves using the cross product on vectors CA and CB.

Find the vectors: CA [1, 1] CB [-3, 0] Calculate the determinant: A (1/2) * |1 -3 1 0| (1/2) * (1 * 0 - (-3) * 1) 1.5

For a visual understanding, you can draw a rough sketch of the triangle, and plot the points on a coordinate system.

Using these methods, we can consistently find the area of the triangle ABC, which is 1.5 square units (or 3/2 square units).

It's worth noting that there are multiple ways to solve this problem, and each method provides a different perspective and can be useful in different situations. Familiarity with these methods will greatly enhance your geometric problem-solving skills.