Problem 45 - College Algebra

Question

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices\(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work. Use determinants to find the area of the triangle whose vertices are \((3,-5),(2,6),\) and \((-3,5)\).

Step-by-Step Solution

Answer

The area of the triangle is 28

Step 1: Setting the Determinant

Set up the determinant matrix from the coordinates of the three vertices (3,-5), (2,6), and (-3,5) as below: \[ \begin{vmatrix}3 & -5 & 1 \\ 2 & 6 & 1 \\ -3 & 5 & 1 \end{vmatrix} \]

Step 2: Evaluating Determinant Performing Operation

Evaluate the determinant by performing two operations - cross multiplication over the diagonals and subtraction. Simplify this to the following form: \[ \begin{vmatrix}3 & -5 & 1 \\ 2 & 6 & 1 \\ -3 & 5 & 1 \end{vmatrix} = 3 \cdot 6 \cdot 1 + -5 \cdot 1 \cdot -3 + 1 \cdot 2 \cdot 5 - 1 \cdot 6 \cdot -3 - 1 \cdot -5 \cdot 2 - 3 \cdot 1 \cdot 5 \]

Step 3: Calculating the Determinant Value

Calculate the determinant value: \[ 18 + 15 + 10 + 18 + 10 - 15 = 56 \]

Step 4: Applying the Area Formula

Substitute the determinant value into the formula for the area of a triangle and calculate the final area result: \[ Area = \(\pm\frac{1}{2}\) \cdot |\text{determinant}| = \frac{1}{2} \cdot 56 \]

Key Concepts

Area of a Triangle

Understanding how to calculate the area of a triangle using determinants is an insightful approach. Normally, the area of a triangle can be found through formulas involving base and height or using Heron's formula. However, when the vertices of a triangle are given in terms of coordinates, determinants offer a handy solution.
The formula for determining the area, in this context, is derived from the determinant matrix of a 3x3 grid, which includes the coordinates of the vertices and additional values:

• The first two columns represent the x and y coordinates of the triangle's points.
• The third column consists of ones, which sets up the matrix for computation.

The determinant then helps in computing the area by considering only the values of the coordinates. Interestingly, the formula for the area extracted from the determinant takes the following form:\[\text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \x_{2} & y_{2} & 1 \x_{3} & y_{3} & 1\end{array}\right|\]It is essential to select the appropriate sign to ensure a positive area result. This approach can save time and offers mathematical elegance.

Rectangular Coordinate System

The rectangular coordinate system, sometimes known as the Cartesian plane, is foundational in understanding how to plot and use coordinate points in geometry. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
This system allows us to describe locations of points using ordered pairs, each pair corresponding to a position on the plane:

• The first number in the pair indicates the distance along the x-axis.
• The second number corresponds to the distance along the y-axis.

This system not only helps in graphing functions and plotting shapes but also in practical situations where precise locations are needed.
Understanding this coordinate system is crucial for laying out the vertices of a triangle when using determinant methods to find the area, as was the case for the given vertices \((3,-5), (2,6),\) and \((-3,5)\). Determining the area by setting up these points within the grid allows for a visual representation that supports mathematical calculations.

Matrix

The concept of a matrix is central to many areas of mathematics, including the calculation of areas using determinants. A matrix is essentially a collection of numbers arranged in rows and columns, similar to a grid.
Matrix notation has widespread applications, from solving linear equations to encoding geometric transformations.In this specific context, a 3x3 matrix is used to organize the coordinates of a triangle's vertices, along with additional constant values:

• The first row includes the coordinates of the first vertex \((x_1, y_1, 1)\).
• The second row includes the coordinates of the second vertex \((x_2, y_2, 1)\), and so on.
• This setup allows for streamlined operations and simplifies the process of finding the determinant.

These matrices are particularly useful in visualizing and computing problems involving multiple steps, such as determining the area of a triangle by using algebraic techniques.

Cross Multiplication

Cross multiplication is a calculation technique often employed within algebra when working with determinants. It involves multiplying numbers on a diagonal and summing or subtracting specific products, providing an efficient means of handling more complex mathematical expressions.

• Within the determinant matrix used for area calculation, cross multiplication helps streamline operations and simplifies steps.
• The process involves moving through different diagonals and combining products.

For instance, to evaluate the given determinant matrix for the triangle with vertices \((3,-5), (2,6), (-3,5)\), we perform the following cross multiplication operations:

• Calculate products: \(3 \cdot 6 \cdot 1, -5 \cdot 1 \cdot -3, 1 \cdot 2 \cdot 5\)
• Subtract: \(1 \cdot 6 \cdot -3, 1 \cdot -5 \cdot 2, 3 \cdot 1 \cdot 5\)

Cross multiplication simplifies the process, reducing the likelihood of arithmetical errors and enabling swift determinations of matrix values critical to areas and other mathematical tasks.