Problem 42 - 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. $$P(2,-2), Q(0,0), R(-3,-4)$$

Step-by-Step Solution

Answer

The area of the triangle is 7 square units.

Step 1: Determine the Coordinates

Identify the coordinates of the triangle vertices: \( P(2, -2) \), \( Q(0, 0) \), and \( R(-3, -4) \). These are \( (x_1, y_1) = (2, -2) \), \( (x_2, y_2) = (0, 0) \), and \( (x_3, y_3) = (-3, -4) \).

Step 2: Set up the Determinant Matrix

Set up the matrix for the determinant using the coordinates: \[\begin{bmatrix}2 & -2 & 1 \0 & 0 & 1 \-3 & -4 & 1\end{bmatrix}\]

Step 3: Calculate the Determinant

Compute the determinant of the matrix: \[\text{det} = 2 \cdot (0 \cdot 1 - 1 \cdot (-4)) - (-2)(0 \cdot 1 - 1 \cdot (-3)) + 1(0 \cdot (-4) - 0 \cdot (-3))\]\[= 2 \cdot 4 - (-2) \cdot 3 + 0\]\[= 8 + 6 + 0\]\[= 14\]

Step 4: Calculate the Area

Use the formula for the area of the triangle: \( \text{Area} = \frac{1}{2} |D| \). Here, \( D = 14 \). So the area is: \[\text{Area} = \frac{1}{2} \times |14| = 7\]

Key Concepts

Triangle Coordinate Geometry

Understanding triangle coordinate geometry involves determining the properties of a triangle using the coordinates of its vertices. In this context, a triangle is defined by three points in a plane, known as its vertices. Each vertex has a pair of coordinates\((x, y)\), which provides a precise location on the coordinate plane.
To analyze triangles in the coordinate plane, we often start by identifying the position of each vertex using given coordinate points. For example, in the exercise above, the vertices of triangle\( PQR \) are located at coordinates \((2, -2)\),\((0, 0)\), and \((-3, -4)\). Once the vertices are known, we can apply various mathematical techniques, like using determinants, to calculate important properties like area.
Working with coordinates makes it easier to deal with transformations and understand relative positions, such as whether the triangle is oriented clockwise or counterclockwise. This foundational concept is essential in geometry and serves as a basis for more advanced topics in mathematics.

Area of Triangle

The area of a triangle defined by Cartesian coordinates can be calculated using a formula that involves a determinant. This method is particularly useful when the triangle's vertices are known, as it provides a straightforward calculation of its area.
The determinant formula for finding the area requires setting up a 3x3 matrix using the coordinates of the vertices. Each row of the matrix represents a vertex, with columns corresponding to the x-coordinate, y-coordinate, and the number 1. For example:

• Vertex \( (2, -2) \)
• Vertex \((0, 0)\)
• Vertex \((-3, -4)\)

Setting these vertices into our matrix equation provides an easy calculation. The determinant is then calculated by cross-multiplying and subtracting according to specific rules.
The area is half of the absolute value of this determinant. In our example, the calculated determinant was \(14\). Therefore, the area of triangle \(PQR\) is \(\frac{1}{2} \times |14| = 7\). This process highlights the interplay between algebraic methods and geometric concepts in solving real-world problems.

Linear Algebra

Linear algebra serves as a key underpinning in solving geometric problems involving triangles, notably in the use of matrices and determinants. A fundamental tool in linear algebra is the determinant, which can be used to find the area of polygons like triangles.
In the context of the problem above, linear algebra facilitates the calculation of a triangle's area through the determinant of a matrix formed by the vertices' coordinates. This matrix transformation is not just limited to 2D space, but extends to various applications in higher dimensions.
Linear algebra offers concepts and methods that simplify complex geometric calculations, enabling us to handle transformations, solve systems of equations, and interpret the properties of various geometrical shapes. Understanding how to set up and manipulate matrices is not only essential in mathematics but also in fields such as computer graphics, physics, and engineering. In this exercise, linear algebra has allowed us to precisely calculate the triangle's area with relative ease, demonstrating the power and versatility of these mathematical techniques.