Problem 41 - University Calculus: Early Transcendentals

Question

Find the areas of the triangles whose vertices are given. $$A(0,0), \quad B(-2,3), \quad C(3,1)$$

Step-by-Step Solution

Answer

The area of the triangle is 5.5 square units.

Step 1: Understanding the Problem

We need to find the area of a triangle whose vertices are given by the points \(A(0,0)\), \(B(-2,3)\), and \(C(3,1)\). We'll use the formula for the area of a triangle given vertices with coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\).

Step 2: Write Down the Area Formula

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

Step 3: Substitute the Coordinates into the Formula

Substitute the given points \(A(0,0)\), \(B(-2,3)\), \(C(3,1)\) into the formula:\[ ext{Area} = \frac{1}{2} \left| 0(3-1) + (-2)(1-0) + 3(0-3) \right|\]

Step 4: Simplify the Expression

Calculate the expression inside the absolute value:\[0(3-1) = 0, \quad (-2)(1-0) = -2, \quad 3(0-3) = -9\]So the expression becomes:\[ ext{Area} = \frac{1}{2} \left| 0 - 2 - 9 \right| = \frac{1}{2} \left| -11 \right|\]

Step 5: Calculate the Absolute Value and the Final Area

The absolute value of \(-11\) is \(11\). Now, multiply by \(\frac{1}{2}\):\[\frac{1}{2} \times 11 = 5.5\]Therefore, the area of the triangle is \(5.5\) square units.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry that uses a coordinate system to explore geometric problems.
In this system, every point in the plane is identified through a pair of numerical coordinates. These coordinates define the location of a point in a two-dimensional space.

• The first number is usually the x-coordinate, representing the horizontal position.
• The second number is the y-coordinate, representing the vertical position.

This simple idea serves as the foundation for complex geometric calculations, including calculating distances, slopes, and areas of various shapes.
By using coordinate geometry, we can easily manage and manipulate points, lines, and shapes using algebraic equations.
This approach is incredibly useful in finding the area of triangles when the vertices have specified coordinates.

Triangle Vertices

In geometry, a triangle is a polygon with three edges and three vertices.
When finding the area of a triangle using coordinates, it is crucial to understand what vertices and how they contribute to calculations.
Each vertex of the triangle is identified by an ordered pair of numbers in the coordinate plane.
In your exercise, you are given three points:

• A with coordinates egin{math}(0, 0) uildrel{ ext{(point of origin - easiest to work with)}} ight.)
• B with coordinates egin{math}(-2, 3) uildrel{ ext{(a point in the second quadrant)}} ight.)
• C with coordinates egin{math}(3, 1) uildrel{ ext{(a point in the first quadrant)}} ight.)

These vertices form the triangle that we aim to analyze and work with in the coordinate plane to determine the area.

Area Calculation

Calculating the area using coordinates involves a specific mathematical formula that simplifies the process.
The formula for finding the area of a triangle with given vertices, expressed as egin{math}(x_1, y_1), (x_2, y_2), (x_3, y_3) ight., egin{math} ext{is:} ightrace rac{1}{2} imes| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|.
Using this formula, we apply our coordinates:

• egin{math} (0,0) ight.)
• egin{math}(-2,3) ight.)
• egin{math}(3,1) ight.)

The calculation steps involve substituting these into the formula and simplifying:

• The expression becomes begin{math}0(3-1) + (-2)(1-0) + 3(0-3). ight.) - This simplifies to: 0, -2, -9.
  
• The sum is egin{math}-11 ight. ight.)

Find the absolute value egin{math}11, ight. ight.) then calculate the area as: egin{math}rac{1}{2} imes 11, ight.) ight.) Finally, the area is egin{math}5.5 ight.) ext{square units, which is}.
With understanding of the logic and calculation practiced, determining the area of triangles using vertices can enhance problem-solving skills in geometry.