Problem 43 - Thomas Calculus

Question

Triangle area Find a formula for the area of the triangle in the \(x y\) -plane with vertices at \((0,0),\left(a_{1}, a_{2}\right),\) and \(\left(b_{1}, b_{2}\right) .\) Explain your work.

Step-by-Step Solution

Answer

The area is \( \frac{1}{2} |a_1b_2 - b_1a_2| \).

Step 1: Understanding the Triangle in the Coordinate Plane

We are given a triangle with vertices at \((0, 0)\), \((a_1, a_2)\), and \((b_1, b_2)\). The task is to find the formula for the area of this triangle. We can visualize the vertices as points on the coordinate plane.

Step 2: Use the Determinant Formula for Area

To find the area of a triangle given its vertices in the coordinate plane, we use the formula derived from the determinant of a matrix containing the vertices. The area \(A\) is given by:\[A = \frac{1}{2} \left| a_1(b_2 - 0) + b_1(0 - a_2) + 0(a_2 - b_2) \right|\]This simplifies to: \[A = \frac{1}{2} \left| a_1b_2 - b_1a_2 \right|\]

Step 3: Apply the Determinant Formula

By substituting the coordinates into the determinant-based formula, we compute:\[A = \frac{1}{2} \left| a_1b_2 - b_1a_2 \right|\]This gives us the area of the triangle defined by the given vertices in terms of \(a_1, a_2, b_1,\) and \(b_2\).

Key Concepts

Determinant Formula

The determinant formula is a powerful tool in mathematics, especially in linear algebra, used for various applications including finding areas of polygons. When computing the area of a triangle with vertices in a coordinate plane, we can use a matrix determinant to simplify the calculation. Here, the determinant helps express the area of the triangle without having to draw it or perform complex geometric operations.

The formula for the area using the determinant is:

• Take the coordinates of the vertices:
  • (0, 0),
  • (a_1, a_2),
  • (b_1, b_2).
• Substitute the coordinates into the area formula:
• Use: \[ A = \frac{1}{2} \left| a_1b_2 - b_1a_2 \right| \]

This formula arises from setting up a matrix with coordinates and finding its determinant, capturing the signed area of the parallelogram formed by the vectors. The absolute value of half of this determinant gives the triangle's area.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the use of coordinates to represent geometric shapes. This approach enables mathematicians and students alike to tackle geometric problems using algebra. By plotting points on a plane using their coordinates, geometric figures such as triangles can be analyzed with precision and ease.

In the case of our triangle:

• We consider its vertices
  • (0, 0),
  • (a_1, a_2),
  • (b_1, b_2).
• These points are plotted on the xy-plane, forming the vertices of the triangle.
• Using these coordinates within the coordinate system allows us to apply algebraic formulas to find measurements, like the area.

This method is not only efficient but also deeply connects algebraic equations to geometric visualization, providing a comprehensive understanding of spatial relationships.

Triangle Vertices

Vertices are the backbone of any polygon, marking the precise points where two sides of the shape meet. In a triangle, these are considered as its corners, anchoring the shape in space. Understanding the importance of vertices helps in visualizing and calculating characteristics like area and perimeter.

In our example, the triangle's vertices are given as:

• (0, 0): This is the origin, simplifying calculations as one point is at an axis intercept.
• (a_1, a_2): This point represents a vertex spaced somewhere within the plane, described by its coordinates.
• (b_1, b_2): Similarly, this vertex complements the other two, completing the triangle.

By acknowledging each vertex's coordinates, we understand their roles in forming the triangle. This understanding is crucial for applying formulas to calculate the desired dimensions, such as area.

Area Calculation

Calculating the area of a triangle using its vertices in the coordinate plane provides a reliable and quick method. The determinant formula, as discussed earlier, helps in getting the area without complex geometric construction. Here’s how the calculation unfolds:

• Identify the triangle's vertices as coordinates.
• Plug these into the determinant-based area formula.
• Use the formula: \[ A = \frac{1}{2} \left| a_1b_2 - b_1a_2 \right| \]
• This formula simplifies to give the absolute area, ensuring no negative values are considered, since the area is always a positive quantity.

By following these steps and inserting the actual values, you can easily find the triangle's area, making this method particularly useful for students and engineers alike, who often work with coordinate systems.