Question

Use a determinant to find the area of the triangle with the given vertices. $$ (-2,4),(2,3),(-1,5) $$

Short answer

The area of the triangle is 2.5 square unit.

Step-by-step solution

Identify the coordinates

First, we need to identify the coordinates of the points. These are given as (-2,4) for point A, (2,3) for point B, and (-1,5) for point C. Therefore, x1=-2, y1=4, x2=2, y2=3, x3=-1, y3=5.

Substitute the values into the formula

Substitute the values into the formula for area of a triangle using determinants. |Area of Triangle| = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. So the absolute value of the area becomes |Area of Triangle| = 1/2 * |-2(3 - 5) + 2(5 - 4) -1(4 - 3)|.

Calculate the area

Compute the values inside the formula. |Area of Triangle| = 1/2*|-2(-2) + 2*1 -1*1| = 1/2*(4 + 2 - 1) = 1/2*5 = 2.5 square unit.

Key concepts

Area of a Triangle

Finding the area of a triangle using determinants is a neat trick from linear algebra that simplifies calculations. This method involves using the coordinates of the vertices of the triangle. Here, our vertices are given as points \((-2,4)\), \((2,3)\), and \((-1,5)\). By using a specific determinant formula, we can compute the area effectively.

The formula for the area of a triangle using determinant is:

• \( \text{Area} = \frac{1}{2} \times | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \)

This formula works by calculating a matrix determinant derived from the coordinates of the vertices:

• \(x_1, y_1\) are the coordinates of the first vertex.
• \(x_2, y_2\) are the coordinates of the second vertex.
• \(x_3, y_3\) are the coordinates of the third vertex.

The result gives twice the area of the triangle, thus dividing by 2 gives the area itself.

This method is advantageous because it allows computation without needing to draw a diagram or apply trigonometry. It simplifies situations where points are difficult to visualize or not aligned along axes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates and formulas to represent and analyze geometric figures. In our exercise, we dealt with the triangle in a coordinate plane, using the vertices \((-2, 4)\), \((2, 3)\), and \((-1, 5)\).

With coordinate geometry, every point is defined by a pair of numerical coordinates:

• The first number represents the point's position along the x-axis (horizontal).
• The second number indicates the position along the y-axis (vertical).

When applied to shapes like triangles, coordinate geometry allows us to calculate lengths, areas, and other properties by applying algebraic techniques. This expands the possibilities for solving geometric problems that would otherwise require synthetic methods like compass and straightedge constructions.

In the provided exercise, coordinate geometry allowed us to swiftly find the area using determinants. It provided a clear and systematic way to handle vertices, turning what could be an abstract calculation into a concrete process with algebraic expressions.

Matrix Calculations

Matrix calculations are an essential component of linear algebra. They involve working with matrices, which are arrays of numbers arranged in rows and columns. In this exercise, a matrix determinant was used to find the area of a triangle.

A determinant is a particular value that can be computed from a square matrix. For a 2x2 matrix:

• If \(\begin{matrix} a & b \ c & d \end{matrix}\), then the determinant is calculated as \(ad - bc\).

When applied to geometry, determinants are useful because they can represent linear transformations and changes in area. In our exercise, the determinant approach simplifies finding the area by using an easy formula instead of relying on traditional geometry methods.

Understanding matrix calculations expands your mathematical toolset, providing ways to solve problems across different areas of mathematics, physics, and engineering. It's this versatility that makes matrices and determinants especially valuable for students to learn.