Problem 46 - Algebra and Trigonometry

Question

In Exercises 45–46, find the area of the triangle with the given vertices. Round to the nearest square unit. $$ (-2,-3),(-2,2),(2,1) $$

Step-by-Step Solution

Answer

The area of the triangle with vertices (-2,-3),(-2,2), and (2,1) is 9 square units.

Step 1: Substitute the Coordinates

Recognize the ordered pairs of the vertices of the triangle in the formula for the area of a triangle given by coordinates. The formula is \(A = 0.5 \times |x_1(y_2 − y_3) + x_2(y_3 − y_1) + x_3(y_1 − y_2)|\). The given vertices are (-2,-3),(-2,2), and (2,1), so \(x_1 = -2, y_1 = -3, x_2 = -2, y_2 = 2, x_3 = 2 \) and \(y_3 = 1\).

Step 2: Calculate the Area

When the known values are substituted into the formula, the calculation of the area \(A\) is \(A = 0.5 \times |-2(2 - 1) + -2(1 - -3) + 2(-3 - 2)|\). This simplifies to \(A = 0.5 \times |-2(1) + -2(4) + 2(-5)| = 9\).

Step 3: Round to the nearest square unit

Given that the area, 9, is already rounded to the nearest square unit, we can conclude at this point.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe and analyze points, lines, and shapes on a coordinate plane. The coordinate plane, with its x and y-axes, allows for a visual and algebraic representation of geometric figures.

When it comes to triangles, coordinate geometry can be particularly useful. By representing the vertices of a triangle as ordered pairs of coordinates, we can apply algebraic methods to solve geometric problems, such as calculating the area of the triangle, without using traditional geometric constructions. It's also easier to work with coordinates because it involves familiar arithmetic and algebraic operations.

Triangle Area Formula

To find the area of a triangle given its vertices' coordinates, a specific formula can be used, which is derived from the determinant of a matrix in linear algebra. This formula is expressed as:
\[ A = 0.5 \times |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]

Where \(A\) represents the area of the triangle, and \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices. The absolute value is used to ensure the area is a positive number, as the triangle's orientation on the plane could result in a negative value from the formula. This formula is a direct application of an area's calculation from coordinate geometry, and it simplifies the process, avoiding the need for other measurements such as base and height.

Vertex of a Triangle

The vertex of a triangle is one of the three corners or intersections of the triangle's sides. In a coordinate plane, each vertex is represented by an ordered pair of numbers, which denote its position along the x (horizontal) and y (vertical) axes. For example, the points \((-2, -3)\), \((-2, 2)\), and \((2, 1)\) in our exercise represent the three vertices of the triangle.

The vertices play a crucial role in coordinate geometry as they serve as the foundational points from which the properties of the triangle, such as its area, perimeter, and angles, can be calculated using various algebraic formulas. Understanding the vertex concept allows for easier manipulation of geometric figures within a coordinate system.