Problem 42 - Thomas Calculus

Question

Find the areas of the triangles whose vertices are given in Exercises \(41-47 .\) $$ A(-1,-1), \quad B(3,3), \quad C(2,1) $$

Step-by-Step Solution

Answer

The area of the triangle is 2 square units.

Step 1: Understanding Triangle Area Formula

To find the area of a triangle with vertices at coordinates \(A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)\), you can use the formula: \[\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|.\]We'll plug the coordinates of the points A, B, and C into this formula.

Step 2: Plugging in Coordinates

Substitute the coordinates of points \(A(-1,-1), B(3,3), C(2,1)\) into the triangle area formula:\[\text{Area} = \frac{1}{2} \left| (-1)((3)-(1)) + (3)((1)-(-1)) + (2)((-1)-(3)) \right|\]

Step 3: Simplify the Expression

Now simplify the expression step by step:1. Calculate each term separately: - \((-1)((3)-(1)) = -1 imes 2 = -2\) - \(3((1)-(-1)) = 3 imes 2 = 6\) - \(2((-1)-(3)) = 2 imes (-4) = -8\)2. Combine these results: \(-2 + 6 - 8\)

Step 4: Evaluate the Absolute and Calculate Area

Combine the results from Step 3 to get:\[-2 + 6 - 8 = -4.\]Take the absolute value and multiply by 1/2:\[\text{Area} = \frac{1}{2} \times |-4| = \frac{1}{2} \times 4 = 2.\]

Step 5: Conclusion

The area of the triangle with vertices at A(-1,-1), B(3,3), and C(2,1) is 2 square units.

Key Concepts

coordinate geometry

Coordinate geometry is a branch of mathematics that uses coordinate points to represent geometric figures. It connects algebra and geometry through graphs and equations.
For triangles, the vertices are the key points where sides meet. By assigning numerical values to these points, we can analyze the properties of shapes and solve geometrical problems efficiently.
Imagine a rectangular grid where each point is identified by a pair of numbers, called coordinates. Using this system, you can pinpoint exact locations in a plane. This precise method of representation makes it easier to calculate and derive various geometric properties, such as finding the area of polygons.

• The x-coordinate reflects the position along the horizontal axis.
• The y-coordinate reflects the position along the vertical axis.
• Each point on a plane is represented as \(x, y\).

triangle vertices

A triangle, one of the simplest polygons, is defined by its three vertices. These points are essentially the corners of the triangle and determine its shape and size.
The positions of these points can be described using ordered pairs in coordinate geometry, which are used in various calculations, like finding the triangle's area.
Knowing the coordinates of the vertices is critical because the calculations to find properties such as area, perimeter, or centroid depend on these values. Let's break it down further:

• Vertex A \(x_1, y_1\)
• Vertex B \(x_2, y_2\)
• Vertex C \(x_3, y_3\)

Each vertex contributes to the overall dimensions of the triangle, and different arrangements can lead to various types of triangles, such as scalene, isosceles, or equilateral.

area formula for triangles

In coordinate geometry, the area of a triangle can be easily calculated using a specific formula involving its vertices' coordinates. This formula is preferred in many cases because it only requires basic arithmetic.
The formula is: \[ ext{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]. This might seem complex at first, but breaking it down step-by-step can simplify its application.
To find the area:

• Substitute each set of \(x, y\) coordinates of the triangle's vertices into the formula.
• Perform arithmetic operations for each term in the equation.
• Sum up the computed values and take the absolute value.
• Finally, multiply by \(\frac{1}{2}\) to get the area in square units.

This method effectively leverages the coordinate plane to compute areas, streamlining the process of dealing with complex vertex arrangements. It's a powerful tool, especially when dealing with non-conventional shapes where traditional geometric formulas are less straightforward.