Problem 57 - Precalculus: Mathematics for Calculus

Question

Sketch the triangle with the given vertices, and use a determinant to find its area. $$(0,0),(6,2),(3,8)$$

Step-by-Step Solution

Answer

The area of the triangle is 21 square units.

Step 1: Plot the Triangle

Begin by plotting the given vertices on a coordinate plane. The vertices are A(0,0), B(6,2), and C(3,8). Draw lines to connect these points, forming a triangle.

Step 2: Set Up the Determinant

Use the formula for the area of a triangle using vertices at \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) given by \[Area = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix} \right|.\] Substitute each point coordinate into the determinant:\[Area = \frac{1}{2} \left| \begin{vmatrix} 0 & 0 & 1 \ 6 & 2 & 1 \ 3 & 8 & 1 \end{vmatrix} \right|.\]

Step 3: Calculate the Determinant

Calculate the value of the determinant:\[\begin{vmatrix} 0 & 0 & 1 \ 6 & 2 & 1 \ 3 & 8 & 1 \end{vmatrix} = 0\times(2\cdot1 - 1\cdot8) - 0\times(6\cdot1 - 1\cdot3) + 1\times(6\cdot8 - 3\cdot2).\]Simplify the expression:\[= 0 - 0 + (48 - 6) = 42.\]

Step 4: Calculate the Area

Substitute the determinant result back into the area formula:\[Area = \frac{1}{2} \times |42| = \frac{42}{2} = 21.\]Thus, the area of the triangle is 21 square units.

Key Concepts

Determinant Calculation

Understanding how to calculate a determinant is vital in coordinate geometry, especially when finding the area of a triangle formed by points on a plane. A determinant, in this context, is a special number calculated from a matrix of numbers. For a triangle with vertices at \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), you can use the formula:

• \[Area = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix} \right| \]

This formula uses a 3x3 matrix that includes the coordinates of the vertices, with an additional column of ones. By finding the determinant of this matrix, you can determine the area enclosed by the triangle. Remember that when you calculate a determinant, you'll multiply and add/subtract combinations of the elements in the matrix, which can be a bit tricky at first.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry through graphs and coordinates. This method is incredibly useful because it allows us to calculate geometric properties using algebraic equations.
Given the points of a triangle, like A(0,0), B(6,2), and C(3,8), coordinate geometry provides us the tools to easily visualize and calculate area, distances, and other properties. It involves plotting the points on a coordinate plane, which is essentially a two-dimensional surface with horizontal and vertical axes. We find our points by moving horizontally and vertically from a starting origin point (0,0).
This approach not just helps in visualizing shapes but also offers a straightforward way to apply mathematical operations such as calculating determinants to find areas, distances between points, or angles.

Triangle Vertices

The term "triangle vertices" refers to the three corners or points where the sides of a triangle meet. In coordinate geometry, these points are represented by ordered pairs such as \((0,0)\), \((6,2)\), and \((3,8)\) in our example.

• When dealing with triangle vertices, it is essential to accurately locate them on a coordinate plane.
• Once plotted, these vertices are connected by straight lines to form the triangle.
Triangles are basic geometric shapes, and understanding their vertices help form the foundation for further study in geometry and mathematics. By examining the arrangement of vertices, not only do we study the shape’s physical dimensions, but we also can compute important properties like area through mathematical principles.

Area Calculation

Calculating the area of a triangle using determinants involves a neat application of algebra. After plotting the triangle's vertices on a coordinate grid, the area calculation can be facilitated using the determinant of a 3x3 matrix as previously discussed.
Here's a simple breakdown of the process:

• Set up the matrix with the vertex coordinates and a column of 1s: \(\begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix}\).
• Calculate the determinant value. This involves taking the difference of products derived from the diagonal elements of the matrix.
• Multiply the absolute value of this determinant by \(\frac{1}{2}\) to get the area.

This method not only provides the exact area but enforces a structured approach to calculations. Using determinants simplifies what could be complex multi-step arithmetic into a straightforward algebraic task.