Problem 49 - 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 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 12 square units.

Step 1: Setup the determinant

Insert the coordinates of the vertices \((3,-5),(2,6),\) and \((-3,5)\) into the formula for the area of a triangle. The determinant will be:\[\begin{pmatrix}3 & -5 & 1 \2 & 6 & 1 \-3 & 5 & 1\end{pmatrix}\]

Step 2: Calculate the determinant

Calculate the determinant by using properties of determinants:\[\frac{1}{2}\left|\begin{array}{ccc}3 & -5 & 1 \2 & 6 & 1 \-3 & 5 & 1\end{array}\right| = \frac{1}{2}\left|3 \cdot 6 \cdot 1 + 2 \cdot 5 \cdot 1 + 1 \cdot 1 \cdot (-5) - 1 \cdot 6 \cdot (-3) - 1 \cdot 2 \cdot 5 - 1 \cdot 3 \cdot 1\right| = 12 \text{ units}^2\]

Key Concepts

Area of a Triangle Using Determinants

Calculating the area of a triangle in the context of algebra and geometry can be elegantly achieved using determinants. Given a triangle with vertices at \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \), the area \( A \) can be found using the formula
\[ A = \frac{1}{2} \bigg| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 darray} \bigg| \]
This method is not just a formula, but it is a powerful application of matrix operations. When you place the x and y coordinates of each vertex into a matrix and find its determinant, you are essentially using the properties of the determinant that link algebraic operations to geometric interpretations. The absolute value ensures that the area is always positive, and the factor of \( \frac{1}{2} \) accounts for the fact that the determinant represents the area of a parallelogram formed by the vectors in two-dimensional space. For our given vertices \( (3,-5), (2,6), (-3,5) \), we set up the determinant and process it to yield the triangle's area, which calculates to 12 square units.
This technique is efficient and avoids the need for more cumbersome geometric calculations. However, remember to always choose the correct sign to ensure the area is positive, as the determinant can return negative values depending on the order of the points.

Properties of Determinants

Determinants play a critical role in various areas of mathematics, including solving systems of linear equations and finding the area of geometric shapes. There are several properties of determinants that make them a powerful tool:

• Linearity in rows and columns: Determinants are linear functions with respect to the rows and columns. This means that if you multiply a row or column by a scalar, the determinant is multiplied by that scalar.
• Zero row/column: If all elements of a row or column are zeros, the determinant of the matrix is zero.
• Elementary row operations: Switching two rows changes the sign of the determinant, adding multiples of one row to another doesn't alter the determinant, and multiplying a row by a constant multiplies the determinant by that constant.
• Triangular matrices: The determinant of a triangular matrix (upper or lower) is equal to the product of the diagonal elements.

When solving for the area of a triangle using determinants, as in the example given, these properties become especially handy. They allow us to manipulate the determinant in a step-by-step manner to extract the desired solution. The algebraic simplicity behind calculating the determinant is what makes it an elegant and preferred approach in various applications.

Rectangular Coordinate System Geometry

In the context of geometry, the rectangular coordinate system, also known as Cartesian coordinates, provides a way to describe the location of points in a two-dimensional space. It consists of two number lines, the x-axis and y-axis, that intersect at a right angle at the origin. Every point in the plane can be located by an ordered pair of numbers \( (x, y) \) that correspond to its horizontal and vertical positions, respectively.
Using this system, geometric shapes such as lines, circles, and polygons can be represented algebraically and analyzed in detail. For instance, the coordinates of the vertices of a triangle can be used to compute distances, slopes, and areas. The rectangular coordinate system simplifies the process of performing geometric calculations, such as finding the area of a triangle using determinants, by providing a standard framework for turning geometric problems into algebraic ones.
Applying this system makes understanding the relationships between algebraic equations and geometric figures more intuitive. When we translate the vertices of a triangle into a matrix to find its area using determinants, we are effectively bridging the gap between algebraic expressions and the geometry of the shape within this coordinate plane.