Problem 58 - Algebra and Trigonometry

Question

Sketch the triangle with the given vertices, and use a determinant to find its area. $$ (-2,5),(7,2),(3,-4) $$

Step-by-Step Solution

Answer

The area of the triangle is 33 square units.

Step 1: Set Up the Matrix for the Determinant

To find the area of a triangle using its vertices, we first set up a 3x3 matrix using the vertices. The vertices are given as \((-2,5),(7,2),(3,-4)\). We add an extra column of 1s to find the area using the determinant method. Thus, the matrix is: \[ \begin{bmatrix} -2 & 5 & 1 \ 7 & 2 & 1 \ 3 & -4 & 1 \end{bmatrix} \]

Step 2: Compute the Determinant

Calculate the determinant of the matrix:\[\text{Det} = -2 \begin{vmatrix} 2 & 1 \ -4 & 1 \end{vmatrix} - 5 \begin{vmatrix} 7 & 1 \ 3 & 1 \end{vmatrix} + 1 \begin{vmatrix} 7 & 2 \ 3 & -4 \end{vmatrix}\]Now, compute each of these 2x2 determinants:\[-2 ((2)(1) - (-4)(1)) = -2 \times 6 = -12 \-5 ((7)(1) - (3)(1)) = -5 \times 4 = -20 \1 ((7)(-4) - (3)(2)) = 1 \times (-34) = -34\]Combine them together: \[ \text{Det} = -12 - 20 - 34 = -66 \]

Step 3: Find the Area from the Determinant

The formula for the area of a triangle using the determinant is given by:\[\text{Area} = \frac{1}{2} \times |\text{Det}|\]Substitute the determinant value we found:\[\text{Area} = \frac{1}{2} \times |-66| = 33\]

Step 4: Conclude with the Final Area

Based on our computation, the area of the triangle with vertices \((-2,5), (7,2), (3,-4)\) is 33 square units.

Key Concepts

Triangle Area

The area of a triangle can be calculated in various ways, and one helpful method is the determinant method. This approach uses the coordinates of the vertices to find the area without needing angles or lengths of sides. Unlike other methods, it works directly with the coordinate points.
To put it simply, when you have a set of three vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), you create a special matrix that helps calculate the area. This relies on the determinant of the matrix which gives twice the area when absolute value and halving are done later. It's a quick and neat trick for using matrices to solve geometric problems with coordinates. Key benefits include:

• Direct usage of vertices, no need for side calculation.
• Simplifies computation for irregular triangles.
• Highly efficient for coordinate geometry tasks.

The determinant method is particularly powerful in analytical geometry tasks, allowing efficient computation even for complex shapes.

Vertices

Vertices can be considered as the fundamental building blocks of any polygon, including triangles. Each vertex in a triangle is a point defined in a plane by its coordinates \(x, y\). For a triangle, you need three such points.
In our exercise, we have three vertices \((-2, 5), (7, 2), (3, -4)\). These serve as corner points. Each pair of coordinates is crucial as it fully dictates one side of the triangle formation. If you visualize these points on a two-dimensional plane, they connect to form the triangle’s three edges.
Understanding vertices and their placement helps recognize shapes and calculate properties like area, as we did with the determinant. These points represent the geometry of the triangle in a very pure form, defining space and shape.

Matrix Determinant

The term 'matrix determinant' might sound complex, but it's a mathematical tool that simplifies many computational tasks. In the context of finding a triangle's area, the determinant provides a mean to calculate it by arranging the triangle's vertices into a special format.
To use the determinant method, we arrange vertices into a 3x3 matrix and fill the last column with 1s. For the vertices \((-2, 5), (7, 2), (3, -4)\), the matrix looks like:
\[ \begin{bmatrix} -2 & 5 & 1 \ 7 & 2 & 1 \ 3 & -4 & 1 \end{bmatrix} \]
The determinant of this matrix is calculated using a formula that involves smaller 2x2 matrices taken from the original 3x3 matrix. Once the determinant is found, its absolute value, divided by 2, gives the triangle's area.
The beauty of the determinant is it translates spatial relationships (like area) into neat algebraic calculations that are efficient to carry out.

Coordinate Geometry

Coordinate geometry or analytic geometry is a unique blend of algebra and geometry, allowing for solving problems using coordinates on planes. It allows us to represent geometric figures using equations—and analyze these relationships through coordinates.
In this exercise, the given vertices serve prominent roles in coordinate geometry. By plotting these points \((-2, 5), (7, 2), (3, -4)\), one visualizes the triangle on the 2D plane.
This method transforms geometry problems into algebraic expressions. Instead of measuring with a ruler, you can solve puzzles about space and shape with precision, manipulating numbers. Techniques like the determinant method showcase how coordinate geometry can provide tools for these calculations using algebraic means, which adds a systematic approach to otherwise visual problems.
Hence, coordinate geometry serves as a powerful bridge between algebra and physical spaces, enabling in-depth analysis and computational ease.