Problem 39 - A Graphical Approach to College Algebra

Question

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(0,0), Q(0,2), R(1,4)$$

Step-by-Step Solution

Answer

The area of the triangle is 1 square unit.

Step 1: Write the Determinant Formula for Area

The area of the triangle is given by the absolute value of the determinant formula: \[ A = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| \] where \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) are the vertices of the triangle.

Step 2: Substitute the Coordinates of Points P, Q, and R

Substitute the coordinates of the points \( P(0,0), Q(0,2), R(1,4) \) into the determinant: \[ \text{det} = \left| \begin{array}{ccc} 0 & 0 & 1 \ 0 & 2 & 1 \ 1 & 4 & 1 \end{array} \right| \]

Step 3: Calculate the Determinant

Calculate the determinant using the formula: \[ \text{det} = 0(2\cdot1 - 1\cdot4) - 0(0\cdot1 - 1\cdot1) + 1(0\cdot4 - 2\cdot1) \] Simplifying, we get \[ \text{det} = 0 - 0 - 2 = -2 \]

Step 4: Compute the Absolute Value and Multiply by 1/2

Compute the absolute value of the determinant and multiply by \( \frac{1}{2} \) to get the area of the triangle: \[ A = \frac{1}{2} \times |-2| = \frac{1}{2} \times 2 = 1 \]

Key Concepts

Area of a Triangle

The area of a triangle with its vertices defined by coordinates can be calculated in various ways. However, an elegant and efficient method uses determinants from linear algebra. This method is particularly useful in coordinate geometry. The formula for finding the area of a triangle with vertices at \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by: \[ A = \frac{1}{2} \left| \text{det} \right| \] where \( \text{det} \) is the determinant of the matrix:

• The top row consists of \((x_1, y_1, 1)\)
• The middle row is \((x_2, y_2, 1)\)
• The bottom row is \((x_3, y_3, 1)\)

This formula uses the properties of matrices and determinants to simplify the calculation of the triangle's area in a straightforward manner.
The absolute value in the formula ensures that the computed area is always positive, reflecting the geometric nature of area as a measure of space.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we describe geometric figures using a coordinate system. This allows for the representation of geometric shapes and their properties algebraically. Using coordinate geometry, we can solve geometric problems by transforming them into algebraic operations.

By placing points on a coordinate plane, we can determine distances, slopes, and other geometrical properties easily. In our case of determining a triangle's area, we use coordinates \( P(0,0), Q(0,2), \) and \( R(1,4) \) to define the vertices of the triangle. These coordinates then feed into the determinant formula to find the area.

• Coordinates help locate each vertex precisely.
• Each coordinate set becomes an input in the formula, making calculations efficient.

Coordinate geometry provides a toolkit that bridges algebra and geometry, turning visual problems into precise algebraic procedures.

Matrix Determinants

Matrix determinants are a fundamental concept in linear algebra. They allow us to perform several useful calculations, such as finding areas, solving systems of equations, and more. A determinant is a special number calculated from a square matrix. In our example, we use a determinant from a 3x3 matrix:

• First row: \((x_1, y_1, 1)\)
• Second row: \((x_2, y_2, 1)\)
• Third row: \((x_3, y_3, 1)\)

The process involves minor expansions and simplifications (as shown in the original solution). The determinant is a sum of products of its elements, each multiplied by a factor of \(-1\) raised to some power.

Calculating the determinant gives a numerical value that when operated upon, provides the area of the triangle in the coordinate plane. This property showcases how matrix operations provide powerful solutions not just in algebra, but in interpreting geometric problems as well.