Question

Use a determinant to find the area of the triangle with the given vertices. $$ (0,-2),(-1,4),(3,5) $$

Short answer

The area of the triangle with the given vertices is 5.5 square units.

Step-by-step solution

Understanding the Formula

The formula to find the area of a triangle when vertices are given is: \[ A = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) | \]. Here (x1, y1), (x2, y2) and (x3, y3) are the vertices of the triangle.

Plug in the given vertices into the formula

Here the coordinates are (0,-2),(-1,4) and (3,5). So, plugging the values into the formula, we get: \[ A = \frac{1}{2} | 0*(4 - 5) - 1*(-2 - 5) + 3*(-2 - 4) | \]

Calculate the Area

Then calculate the area: \[ A = \frac{1}{2} | 0 + 7 - 18 | = \frac{1}{2} * |-11| = \frac{1}{2} * 11 = 5.5 \]. So, the area of the triangle with given vertices is 5.5 square units.

Key concepts

Triangle Area

To determine the area of a triangle using the coordinates of its vertices, we can use a special formula that involves determinants. Determinants help us compute areas efficiently and are particularly useful when given vertices. The formula used is: \[ A = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) | \] This formula calculates the absolute value of a half determinant derived from the vertices' coordinates. This approach takes into consideration the orientation of the triangle's vertices. If vertices are arranged clockwise or counterclockwise, it affects the determinant's sign, hence the use of absolute value.

Vertices

Vertices are key components when it comes to finding the area of a triangle in coordinate geometry. A vertex (plural: vertices) is a point where two edges of a polygon meet. In a triangle, there are three vertices represented by their coordinates in a plane, such as \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\). For our specific problem, the vertices given are \((0, -2)\), \((-1, 4)\), and \((3, 5)\). These points define the triangle on a coordinate plane, and the position of these vertices is crucial for all calculations, including determining the area or any transformation. Understanding how vertices are used in area computations helps in geometric and algebraic applications.

Matrix

A matrix is a powerful tool in mathematics used to organize and simplify computations with multiple numbers, such as vertices. In the context of finding a triangle's area, a matrix is often employed to set up an orderly arrangement of vertex coordinates.Consider the use of a matrix in the determinant formula: it structures the coordinate values to enable their manipulation through algorithms. It is usual to visualize the determinant formula as operating within a matrix-like setup: \[\begin{vmatrix}x_1 & y_1 & 1 \x_2 & y_2 & 1 \x_3 & y_3 & 1\end{vmatrix}\] Though we don't explicitly calculate this determinant in our direct formula used, it helps in understanding the transformation of coordinates which directly support finding the area. The matrix-like setup simplifies transitions through calculations, thanks to leveraging relational properties and scaling factors.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical discipline that uses algebraic equations to represent geometric principles. It involves using a coordinate plane to describe points, lines, and shapes through equations. In finding a triangle's area, coordinate geometry allows us to manipulate vertex positions algebraically. Through coordinate points, we engage in transforming geometric figures into numerical expressions. This transformation is what enables us to apply formulas like the determinant-based area formula, ensuring precision and directness in solving problems like our exercise. Coordinate geometry provides a robust groundwork for exploring geometric properties, making it essential for bridging geometry with algebra. It facilitates spatial insights, helping us visualize and solve complex problems involving distances, slopes, and areas.