Problem 49 - Algebra and Trigonometry

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 Exercises \(49-50\). 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 \(34\) square units.

Step 1: Representation of Points

Assign the points \((3, -5), (2, 6), (-3, 5)\) as \(A(3, -5), B(2,6)\) and \(C(-3, 5)\) respectively.

Step 2: Set Up the Matrix

Using the coordinates of the vertices, construct the \(3 \times 3\) matrix as: \[\left|\begin{array}{ccc} 3 & -5 & 1 \ 2 & 6 & 1 \ -3 & 5 & 1 \end{array}\right|\]

Step 3: Calculation of Determinant

Next step is to calculate the determinant of the matrix, using the formula \[abc + afe + bed - edc - fab - eca\] Therefore, the determinant is calculated as \(3(6.1 + 5.1) - (-5)(2.1 + 1.1) + 1(2.5 + 1.6) - 1(-3.6 + 2.5) - (-5)(-3.1 + 2) + 1(-3.1 - 2.2) = 68\)

Step 4: Calculate Area

Finally, use the formula for the area of the triangle, which is equal to half of the absolute value of the determinant: \[Area = \frac{1}{2} \times |68| = 34\] square units

Key Concepts

Area of a Triangle

Finding the area of a triangle can be done in many ways, but when you have the vertices in a rectangular coordinate system, using determinants is a clever approach. This method ensures precision by converting the problem into calculations you can perform on a small matrix. The formula for the area of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by \(\text{Area} = \pm \frac{1}{2}\left| \begin{array}{lll} x_{1} & y_{1} & 1 \ x_{2} & y_{2} & 1 \ x_{3} & y_{3} & 1 \end{array} \right|\). The plus or minus sign ensures the area is positive, as area cannot be negative. This formula uses the properties of determinants to simplify calculations and is a powerful tool for geometry problems involving coordinates.

Rectangular Coordinate System

The rectangular coordinate system is a two-dimensional system where each point is identified by an ordered pair of numbers, generally written as \((x, y)\). This system is a basic foundation of geometry and helps in representing points graphically. Each coordinate corresponds to the position of the point along the respective axis—the x-axis and y-axis. It provides a way to locate points in space and form geometrical figures like triangles, rectangles, and more. For triangles, having vertex coordinates is crucial as they serve as the input for methods like determinant calculation to find areas. Understanding how to plot and read these points is essential for various mathematical applications.

Matrix Determinant Calculation

The determinant of a matrix is a special number calculated from its elements. For a \(3 \times 3\) matrix, the determinant reveals key properties such as the area of a triangle with given vertices. To find the determinant of this matrix, you apply a specific formula: \(abc + afe + bed - edc - fab - eca\). This combines the multiplication and addition of the elements systematically. Using this process, you get a scalar value which can be used in further calculations. In geometry, especially in problems related to areas, determinants simplify complex computations by reducing them to a few steps.

Vertices of a Triangle

The vertices of a triangle are simply the three points that define its shape. In a rectangular coordinate system, these are expressed as ordered pairs \((x_1, y_1), (x_2, y_2), (x_3, y_3)\). The position of these vertices is crucial as they determine the look and orientation of the triangle. By knowing their coordinates, you can easily use them in various mathematical methods such as calculating the area, determining the centroid, or even finding the lengths of sides. Understanding vertices' roles allows deeper insights into not only triangle geometry but also many other polygonal shapes.