Problem 34 - Calculus: Early Transcendentals

Question

Use calculus to find the area of the triangle with the given vertices. \( (2 , 0) \) , \( (0 , 2) \) , \( (-1 , 1) \)

Step-by-Step Solution

Answer

The area of the triangle is 2 square units.

Step 1: Understanding the Problem

We are given three vertices of a triangle in the coordinate plane: \((2, 0)\), \((0, 2)\), and \((-1, 1)\). Our task is to find the area of this triangle using calculus.

Step 2: Use the Formula for Area of a Triangle

The area \(A\) of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) can be calculated using the determinant formula:\[ A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \].

Step 3: Substitute the Given Coordinates into the Formula

Substitute the vertices \((2, 0)\), \((0, 2)\), \((-1, 1)\) into the formula:\[ A = \frac{1}{2} \left| 2(2-1) + 0(1-0) + (-1)(0-2) \right| \].

Step 4: Calculate Each Term in the Expression

Calculate each term inside the absolute value:- First term: \(2(2-1) = 2\)- Second term: \(0(1-0) = 0\)- Third term: \((-1)(0-2) = 2\)

Step 5: Evaluate the Absolute Value Expression

Add the calculated terms inside the absolute value and compute:\[A = \frac{1}{2} |2 + 0 + 2| = \frac{1}{2} |4| = \frac{1}{2} \times 4 = 2 \].

Step 6: Conclusion

The area of the triangle with the given vertices \((2, 0), (0, 2), (-1, 1)\) is \(2\) square units.

Key Concepts

Area of Triangle

The area of a triangle is a fundamental concept in geometry. It measures the surface covered by the triangle in a two-dimensional plane. When dealing with triangles on a coordinate plane, you can use specific formulas to compute this area. The standard approach involves using base and height, but for triangles with vertices given by coordinates, the determinant formula offers a straightforward path. This method leverages the coordinates of the triangle's vertices, making it particularly useful for problems in coordinate geometry. Identifying the correct formula is crucial for efficient problem-solving in geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric properties. This approach allows for precise definitions and calculations using coordinates. By plotting points in an XY-plane, shapes such as triangles can be analyzed with numerical precision.
In this problem, we are given three points on this plane, each point has an (x, y) form. By calculating the relationships and distances between these points, we can solve geometric problems such as finding the area of a triangle. Coordinate geometry thus merges algebraic methods with geometric applications, simplifying many complex calculations.

Determinant Formula

The determinant formula is a powerful tool in calculus and linear algebra. For finding the area of a triangle, it provides a method that utilizes the coordinates of the triangle's vertices directly. The formula is derived from the concept of determinants in matrices, and it is expressed as:

• \[ A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]

This formula efficiently computes the area by considering the positive or negative sign of the result, ensuring the area value remains positive. Using the determinant formula simplifies calculations immensely, especially when compared to traditional methods requiring base and height.

Vertices

Vertices are the cornerstone in any geometric figure. For a triangle, these are the points where its edges meet. In the context of this exercise, the vertices are given as coordinates:

• \((2, 0)\), \((0, 2)\), and \((-1, 1)\).

Each vertex is a significant part of the triangle's structure, determining its shape and size. By knowing their exact coordinates, we can perform various calculations, such as finding side lengths, angles, and as in this case, the area.
Additionally, the arrangement and selection of these vertices are crucial for solving and understanding geometrical properties in the coordinate plane. They transform abstract geometric problems into solvable mathematical equations.