Problem 39 - Thomas Calculus

Question

Find the areas of the triangles whose vertices are given in Exercises \(39-42\) $$ A(0,0), \quad B(-2,3), \quad C(3,1) $$

Step-by-Step Solution

Answer

The area of the triangle is 5.5 square units.

Step 1: Understand the formula for the area of a triangle

To find the area of a triangle with vertices at \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), we use the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] This formula derives from the determinant of a matrix representing the coordinates of the vertices.

Step 2: Identify coordinates and plug them into the formula

Given the vertices \( A(0,0), B(-2,3), C(3,1) \), identify the coordinates: \(x_1 = 0, y_1 = 0, x_2 = -2, y_2 = 3, x_3 = 3, y_3 = 1\). Plug these into the area formula: \[ \frac{1}{2} \left| 0(3-1) + (-2)(1-0) + 3(0-3) \right| \]

Step 3: Simplify the expression inside the absolute value

Calculate each term separately: 1. \(0 \times 2 = 0\), 2. \(-2 \times 1 = -2\), 3. \(3 \times -3 = -9\). Add these values together: \[0 + (-2) + (-9) = -11\] The expression inside the absolute value becomes \(-11\).

Step 4: Calculate the absolute value and area

Take the absolute value of \(-11\), which is \(11\). Substitute back into the area formula: \[ \frac{1}{2} \times 11 = 5.5 \] Thus, the area of the triangle is \(5.5\) square units.

Key Concepts

Determinant Formula

The determinant formula for finding the area of a triangle using its vertex coordinates is quite intriguing. This method is efficient and leverages the formal determinant from linear algebra. The formula helps in calculating the area of the triangle formed by points

• \( (x_1, y_1) \)
• \( (x_2, y_2) \)
• \( (x_3, y_3) \)

The formula is: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]This stems from the algebraic representation of the area as a matrix determinant. The positions of the coordinates are critical because even a small error in arranging them can lead to inaccurate results. Using determinants, you gain a clear mathematical path to find the area, ensuring precision.To connect the formula to determinants: think of a matrix created from the coordinates, where each column and row combination represents a unique multiplication pair that contributes to the total area expressed through absolute value.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, is the study of geometry using a coordinate system. Here, points are expressed as ordered pairs on a plane. This approach is perfect for solving spatial problems with algebra. When dealing with triangles, each vertex is defined by a pair of coordinates and arranged as \( (x, y) \). The beauty of coordinate geometry is in transforming geometric questions into algebraic equations using these coordinates.The determination of area, distance, and midpoint between points becomes straightforward with coordinate geometry. In our problem, each point such as

• \( A(0,0) \)
• \( B(-2,3) \)
• \( C(3,1) \)

gives a clear pathway to calculate the area or analyze triangle properties relying totally on algebra. This adds convenience and accuracy, especially in more complex triangles positioned on the coordinate plane.

Absolute Value

In mathematics, the absolute value signifies the non-negative value of a number without regard to its sign. Denoted by vertical bars around the number or expression, such as \( |-11| = 11 \), this concept preserves magnitude while ignoring direction.In the context of area calculation, absolute value ensures that the computed area is always a positive measure. This is crucial since a negative area doesn't make sense geometrically. After calculating the determinant value

• \( -11 \)

Due to geometric properties, using the absolute value yields a positive area value, which aligns intuitively with the visual understanding of space covered by a triangle.

Triangles in Mathematics

Triangles are fundamental shapes in mathematics, significantly utilized in various applications from basic geometry to advanced calculus. Defined by three vertices and three sides, triangles are the simplest polygon type but hold immense applicability. Understanding triangles involve examining properties like angles, side-length ratios, and special points (e.g., centroid, orthocenter), which reveal a great deal about geometry and trigonometry principles. In our exercise, calculating the area of a triangle using vertex coordinates highlights the intersection of triangles with coordinate geometry and algebra. It's not just about visualizing a shape on paper; it's about understanding how triangles function within mathematical systems. This base understanding of triangle properties invites further exploration into similar mathematical disciplines, making them essential in engineering, physics, and computer graphics.