Question

A triangle has vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right) .\) The area of the triangle is given by the absolute value of \(D,\) where \(D=\frac{1}{2}\left|\begin{array}{ccc}x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ 1 & 1 & 1\end{array}\right|\) Use this formula to find the area of a triangle with vertices \((2,3),(5,2),\) and (6,5)

Short answer

The area is 5 square units.

Step-by-step solution

Write the Vertices of the Triangle

Identify the coordinates of the vertices of the triangle: \(\left(x_1, y_1\right) = (2, 3)\), \(\left(x_2, y_2\right) = (5, 2)\), and \(\left(x_3, y_3\right) = (6, 5)\).

Set Up the Determinant Formula

The area formula using the determinant is: \[D = \frac{1}{2}\left|\begin{array}{ccc} x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \ 1 & 1 & 1 \end{array}\right|\]

Substitute the Coordinates into the Determinant

Substitute the values of the vertices into the determinant: \[D = \frac{1}{2}\left|\begin{array}{ccc} 2 & 5 & 6 \ 3 & 2 & 5 \ 1 & 1 & 1 \end{array}\right|\]

Calculate the Determinant

Calculate the determinant: \[\left|\begin{array}{ccc} 2 & 5 & 6 \ 3 & 2 & 5 \ 1 & 1 & 1 \end{array}\right| = 2\cdot(2 \cdot 1 - 5 \cdot 1) - 5\cdot(3 \cdot 1 - 5 \cdot 1) + 6\cdot(3 \cdot 1 - 2 \cdot 1)\]

Simplify the Determinant Calculation

Simplify the calculation: \[= 2(2 - 5) - 5(3 - 5) + 6(3 - 2)\] \[= 2(-3) - 5(-2) + 6(1)\] \[= -6 + 10 + 6\] \[= 10\]

Calculate the Area

Using the absolute value of the determinant: \[ \text{Area} = \frac{1}{2} |D| = \frac{1}{2} |10| = 5\]

Key concepts

Triangle Vertices

In coordinate geometry, a triangle is defined by its three vertices. These vertices are simply the corners or points where the sides of the triangle meet. Each vertex is represented by a pair of coordinates \((x, y)\), describing its location on the Cartesian plane. For example, if we have vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), each coordinate pair indicates the exact position of a vertex on the graph. Identify these coordinates correctly, as they are essential in applying formulas for determining the properties of the triangle, such as its area.

Determinant Formula

The determinant formula is a mathematical tool used to find the area of a triangle when its vertices are known. According to this formula, we can form a matrix using the coordinates of the triangle's vertices and then calculate its determinant. The formula is expressed as:

\[ D = \frac{1}{2} \left| \begin{array}{ccc} x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \ 1 & 1 & 1 \end{array} \right| \]

This matrix encapsulates the coordinates of the triangle’s vertices. It is essential to correctly set up this matrix to ensure accurate calculations of the area.

Absolute Value

When computing the area of a triangle using the determinant, taking the absolute value of the result is crucial. This is because determinants can yield negative values, but an area cannot be negative. Using absolute value, we obtain the positive magnitude of the determinant, guaranteeing that the area is non-negative. In the formula \[\text{Area} = \frac{1}{2} \left| D \right|\], the absolute value operation ensures that if the determinant \D\ is negative, we consider its positive counterpart, thus providing the correct measure of the triangle's area.

Coordinate Geometry

Coordinate geometry is a branch of geometry where points are placed on the coordinate plane, described by unique coordinates \((x, y)\). It provides tools to study geometric figures using algebraic equations and calculations. In the context of finding the area of a triangle, coordinate geometry uses the precise locations of vertices on the plane. By using the vertices' coordinates, you can apply the determinant formula to compute the triangle's area. This method is algebraically rich and provides clear, accurate results. It bridges geometry and algebra, allowing for effective and efficient problem solving.