Problem 64 - Precalculus

Question

Apply determinants to find the area of a triangle with vertices, \((-1,-2),(3,4),\) and (2,1)

Step-by-Step Solution

Answer

The area of the triangle is 3 square units.

Step 1: Identify Vertices

First, let's clearly identify the vertices of the triangle. The given vertices are \((-1,-2)\), \((3, 4)\), and \((2, 1)\). These will be used in the formula to find the area using determinants.

Step 2: Formula for Area of Triangle Using Determinants

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

Step 3: Plug Values into Formula

Substitute the coordinates \((-1, -2)\), \((3, 4)\), and \((2, 1)\) into the formula. \[ \text{Area} = \frac{1}{2} \left| (-1)(4-1) + 3(1+2) + 2(-2-4) \right| \].

Step 4: Calculate Each Term

Calculate each term inside the absolute value: 1. \((-1)(4-1) = -3\)2. \(3(1+2) = 9\)3. \(2(-2-4) = -12\)

Step 5: Sum the Terms

Add together all the calculated terms: \(-3 + 9 - 12 = -6\)

Step 6: Compute Absolute Value and Final Area

Take the absolute value and multiply by \(\frac{1}{2}\): \[ \text{Area} = \frac{1}{2} \times |-6| = \frac{1}{2} \times 6 = 3 \].

Key Concepts

area of a triangle

When working with triangles, finding the area is a common task. The area of a triangle gives us the size of the surface enclosed by its three sides. For standard triangles, knowing the base and height is enough. But when dealing with triangles on a coordinate plane, where you know the vertices, you can use a special formula involving determinants.
This formula is particularly useful because it eliminates the need to draw the triangle and calculate its base and height visually. Instead, the determinant offers a straightforward calculation by using the coordinates (vertices) of the triangle's corners.

• It works perfectly for triangles described by their vertices on a plane.
• It's particularly helpful when the vertices include negative coordinates.

The determinant formula connects algebra and geometry and provides a quick and neat way to solve area problems geometrically.

vertices

Vertices are key points on a geometric shape. For triangles, vertices are the three corners where the sides meet. They are represented by coordinates on the Cartesian plane, usually denoted as \((x_1, y_1), (x_2, y_2), (x_3, y_3)\). In the context of determinants, these vertices form the backbone of our calculations.
Using vertices, you don't just calculate the shape's area; you also deeply understand its geometric properties.
Here's how:

• Each vertex represents a unique point on the plane.
• The coordinates tell you both the horizontal and vertical positions of that point.
• These positions help us understand the triangle's overall size and orientation on the plane.

Understanding vertices is crucial for utilizing the determinant formula effectively for calculating the area of triangles.

determinant formula

The determinant formula is an elegant mathematical tool that finds application in numerous areas, including calculating the area of triangles on a coordinate plane.
Specifically, it's used when you know the vertices of the triangle. This formula uses the coordinates of the vertices to provide a quick computation of the area without needing direct measurements.Here's a simple breakdown of the process:

• Identify the coordinates of the three vertices of the triangle: \((x_1, y_1), (x_2, y_2), (x_3, y_3)\).
• Apply the determinant formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \].
• Calculate the values inside the absolute value operation and simplify the expression.
• Take the absolute value of the expression and multiply by \(\frac{1}{2}\) to arrive at the area.

By using this formula, you can very efficiently calculate the area regardless of the triangle's position or orientation on the plane.