Problem 42 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The area of the triangle is 2.

Step 1: Write down the formula for the area of a triangle

The formula for the area of a triangle, given its vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \), is: \[ Area = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]

Step 2: Substitute the given coordinates into the formula

Given the vertices \(A(-1,-1), B(3,3), C(2,1)\), substitute these into the formula: - \(x_1 = -1, y_1 = -1\)- \(x_2 = 3, y_2 = 3\)- \(x_3 = 2, y_3 = 1\)Plug these into the formula:\[Area = \frac{1}{2} \left| (-1)(3-1) + 3(1-(-1)) + 2((-1)-3) \right|\]

Step 3: Simplify the expression inside the absolute value

Calculate the expression inside the absolute value:- \( (-1)(3-1) = -1 \times 2 = -2\)- \(3(1-(-1)) = 3 \times 2 = 6\)- \(2((-1)-3) = 2 \times (-4) = -8\)The expression becomes:\[Area = \frac{1}{2} \left| -2 + 6 - 8 \right|\]

Step 4: Calculate the value inside the absolute value

Combine the terms inside the absolute value:\[-2 + 6 - 8 = -4\]So, the expression inside the absolute values is \(-4\).

Step 5: Apply absolute value and compute the area

Take the absolute value of \(-4\), which is \(4\). Now calculate the area:\[Area = \frac{1}{2} \times 4 = 2\]

Step 6: Conclude with the area value

The area of the triangle with vertices \(A(-1,-1), B(3,3), C(2,1)\) is \(2\).

Key Concepts

Vertices of a Triangle

A triangle has three corners, which we call vertices. Each vertex is a point with an x and y coordinate in a 2-dimensional space. In this exercise, we have three vertices of a triangle, labeled as A, B, and C. Specifically, the vertices are:

• Point A: (-1,-1)
• Point B: (3,3)
• Point C: (2,1)

These points help define the shape and size of our triangle. By using these coordinates, we can calculate different properties of the triangle, such as its area. Remember, the vertices are crucial because they are the starting points for any calculations related to the triangle.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a method that uses algebra to study geometric properties of figures on a coordinate plane. In this context, it allows us to precisely locate points and shapes on a plane using numbers.

Here, the points A, B, and C represent specific locations on the plane, which form a triangle. Coordinate geometry provides tools and formulas, like the one used to calculate areas or distances between points, making it easier to analyze relationships between different shapes.

Understanding coordinate geometry is key for visualizing geometric shapes and solving problems involving them. It's like taking geometry into the world of mathematics, where everything is handled numerically.

Mathematical Formula

To solve for the area of the triangle formed by the vertices A, B, and C, we use a specific mathematical formula. This formula involves the coordinates of the vertices and is expressed as:\[Area = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]
In this formula:

• \(x_1, y_1\) are the coordinates of vertex A.
• \(x_2, y_2\) are the coordinates of vertex B.
• \(x_3, y_3\) are the coordinates of vertex C.

This formula calculates the absolute area without depending on how the triangle is oriented on the plane. Plug in the coordinates for each vertex into the formula to find the area. This powerful formula allows for a quick and accurate calculation of the triangle's area.

Simplifying Expressions

After substituting the coordinates into the area formula, we need to simplify the mathematical expression. The goal here is to calculate each term individually and then combine them effectively:

• First, each of the three main parts of the formula is calculated separately. For example, interpret \((-1)(3-1)\) to get \(-2\), then do the same for other parts like \(3(1-(-1))\) which equals to \(6\), and \(2((-1)-3)\) that equals \(-8\).
• Next, add all these terms together to simplify the expression within the absolute value symbol. Here it simplifies to \(-2 + 6 - 8\) which results in \(-4\).
• Finally, apply the absolute value, turning \(-4\) into \(4\), and multiply by \(\frac{1}{2}\) to get the area \(2\).

Simplifying expressions involves breaking down complex calculations into manageable steps, making it easier to track each part of the problem and verify results. By understanding each step, you can more easily solve similar exercises in the future.