Problem 54 - Precalculus: Mathematical for Calculus

Question

Sketch the triangle with the given vertices and use a determinant to find its area. $$(-2,5),(7,2),(3,-4)$$

Step-by-Step Solution

Answer

The area of the triangle is 33 square units.

Step 1: Understanding the Points

Firstly, identify the vertices of the triangle. The given points are A = (-2, 5), B = (7, 2), and C = (3, -4). These are the vertices of the triangle.

Step 2: Setup the Formula for Area Using Determinants

The area of a triangle given its vertices (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \].

Step 3: Substitute the Vertex Coordinates

Substitute the coordinates into the formula. For A = (-2, 5), B = (7, 2), and C = (3, -4), the formula becomes: \[ \text{Area} = \frac{1}{2} \left| -2(2 - (-4)) + 7((-4) - 5) + 3(5 - 2) \right| \].

Step 4: Simplify Each Term Inside the Determinant

Calculate each part of the expression: \(-2(2 + 4) = -2(6) = -12\), \(7(-4 - 5) = 7(-9) = -63\), \(3(5 - 2) = 3(3) = 9\).

Step 5: Compute the Determinant

Add the results from the previous step: \(-12 - 63 + 9 = -66\).

Step 6: Find the Absolute Value and Divide by 2

Calculate the absolute value: \(|-66| = 66\). Now, divide by 2 to find the area of the triangle: \(\text{Area} = \frac{1}{2} \times 66 = 33\).

Step 7: Sketch the Triangle

Plot the points on a coordinate grid: A at (-2, 5), B at (7, 2), and C at (3, -4). Connect the points with straight lines to form a triangle.

Key Concepts

Triangle Area Formula

The triangle area formula involving determinants is a powerful tool to calculate the area of a triangle quickly and accurately. It relies on the coordinates of the triangle's vertices. The formula is derived from linear algebra and uses the determinant of a matrix that is generated from these coordinates. The formula is expressed as:

• \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Here, \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) are the vertices of the triangle. The determinant measures how the points spread out in the plane, essentially providing the parallelogram's area spanned by two vectors. Taking half of this determinant gives the area of the triangle, as a triangle is half of a parallelogram. The absolute value ensures the area is a positive number. Using this method is particularly efficient for dealing with non-integer and complex coordinates.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry. It's a crucial area of mathematics that manages the position and form of points, lines, and figures in a prescribed coordinate system. The most common system is the Cartesian plane, consisting of a horizontal axis (x-axis) and a vertical axis (y-axis).
In coordinate geometry, everything sits neatly in its place on this grid. You can calculate distances, midpoints, slopes, and areas. Determinants are especially handy when calculating the area enclosed by vertices like in a triangle.

• Given three points, you can ascertain lines between them,.
• Determine various properties such as length and midpoint of these lines.

This interplay between algebra and geometry opens a fantastic way to visualize and solve geometric problems algebraically. It enables not only the calculation of areas but the analysis of various shapes and figures.

Vertices of a Triangle

The vertices of a triangle are the intersection points of its sides, effectively its corner points. In a triangle ABC, the vertices would be labeled as \( A, B, \text{and} C \), each associated with respective coordinates \( (x_1, y_1), (x_2, y_2), \text{and} (x_3, y_3) \).
Understanding these vertices is crucial for various geometric and algebraic calculations:

• They determine the shape and size of the triangle.
• Allow calculations of side lengths using distance formulas.
• Serve as essential inputs for calculating the area via the determinant formula.

When given specific coordinates like \((-2, 5), (7, 2), (3, -4)\), these vertices can be plotted on the Cartesian plane to construct the triangle visually. Moreover, they are foundational in analyzing the properties and attributes of the triangle such as angles, perimeter, and in this case, area. Understanding and plotting vertices enables deeper comprehension of spatial relationships within the coordinate system.