Problem 40 - A Graphical Approach to Precalculus with Limits

Question

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area cqual to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coondinates as given. (GRAPH CAN'T COPY) $$P(0,1), Q(2,0), R(1,5)$$

Step-by-Step Solution

Answer

The area of the triangle is 4 square units.

Step 1: Understand the Determinant Formula

The formula for the area of a triangle using coordinates involves the determinant of a 3x3 matrix. For vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the area is \(\frac{1}{2} \times |D|\) where \[D = \operatorname{det}\begin{bmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{bmatrix}\]

Step 2: Identify geometric figures and given information

Identify shapes, angles, lengths, and relationships.

Step 3: Apply geometric formulas and theorems

Use appropriate formulas and theorems to set up equations.

Step 4: Solve and compute

Solve equations and compute required quantities.

Step 5: State the final answer

Express the answer with correct units.

Step 6: Conclude with the answer

The area of the triangle is 4 square units.

Key Concepts

Area of Triangles

The area of a triangle can be calculated using various methods, one of which is through the use of determinants involving coordinate geometry. When you have the vertices of a triangle given as coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), you can use the formula: \[ \text{Area} = \frac{1}{2} |D| \] The value \(|D|\) is the absolute value of the determinant of a special 3x3 matrix made up of the coordinates of these points.By substituting the coordinates of the triangle's vertices into the matrix:\[ D = \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix} \]You can calculate the value of this determinant, and hence the area of the triangle. This method of finding the area using coordinate points is extremely useful because it leverages the linear algebra concept of determinants, which simplifies the calculation process once you understand how to set up the matrix.

Coordinate Geometry

Coordinate geometry, sometimes referred to as analytic geometry, is the study of geometry using the coordinate system. This branch of geometry enables us to describe geometric figures algebraically and solve geometric problems computationally. Each point in the coordinate plane is defined by an ordered pair of numbers \(x, y\), where \(x\) represents the horizontal position and \(y\) represents the vertical position.For triangles, understanding the vertices in terms of coordinates allows us to explore different properties like length and area. For example, the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be found using the distance formula:\[ \text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \] This forms the basis for many calculations, including those involving the areas where determinants provide a more streamlined solution.Using coordinates lets us harness the power of algebra in solving problems that involve shapes and figures. This approach allows for the flexibility of using different mathematical tools, like matrices and determinants, to find solutions efficiently.

Matrix Operations

Matrix operations, like determinant evaluation, can be a powerful tool for solving geometric problems in coordinate geometry. Matrices are arrays of numbers arranged in rows and columns. In the context of finding the area of a triangle, the determinant of a 3x3 matrix defined by the coordinates of the triangle's vertices plays a central role.For a 3x3 matrix of the form \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \] The determinant is calculated as:\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]When it comes to triangles, having a matrix representation allows us to efficiently calculate areas using predefined formulas. These operations are fundamental in linear algebra, frequently used in computer science, physics, and engineering to solve complex problems.By understanding how to work with matrices and determinants, we open up powerful techniques to solve problems that might otherwise be very complicated using basic geometry alone.