Problem 44 - A Graphical Approach to College Algebra

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 equal 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 coordinates as given. $$P(-1,0), Q(-3,5), R(5,2)$$

Step-by-Step Solution

Answer

The area of the triangle is 17 square units.

Step 1: Formulate the Determinant

Use the given vertices of the triangle, \(P(-1,0), Q(-3,5), R(5,2)\), to set up the determinant needed to calculate \(D\). Arrange the coordinates into the matrix form: \[ \begin{vmatrix} -1 & 0 & 1 \ -3 & 5 & 1 \ 5 & 2 & 1 \end{vmatrix} \]

Step 2: Calculate the Determinant

Expand the determinant \(D\) using the formula for a \(3 \times 3\) determinant: \[ D = x_1(y_2z_3 - y_3z_2) - y_1(x_2z_3 - x_3z_2) + z_1(x_2y_3 - x_3y_2) \]Substituting the entries:\[ = -1(5 \cdot 1 - 2 \cdot 1) - 0(-3 \cdot 1 - 5 \cdot 1) + 1(-3 \cdot 2 - 5 \cdot 5) \]Simplify each term:\[ = -1(5 - 2) + 0 + 1(-6 - 25) \]\[ = -3 - 31 \]\[ = -34 \]

Step 3: Calculate Absolute Value and Area

To find the area using determinant \(D\), compute its absolute value: \[ |D| = | -34 | = 34 \]Divide by 2 to get the area of the triangle:\[ \text{Area} = \frac{1}{2} \times 34 = 17 \]

Key Concepts

Determinant

The determinant is a special number calculated from a square matrix. It plays a crucial role in geometry, especially when finding areas of triangles when the coordinates of vertices are known. For a triangle with vertices at \( \left(x_1, y_1\right), \left(x_2, y_2\right), \left(x_3, y_3\right) \), we use a specific determinant formula:

• Put the vertex coordinates into a 3x3 matrix: \[ \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \ \end{vmatrix} \]
• Calculate the determinant using an expansion method suitable for 3x3 matrices.

The calculated determinant helps in finding the triangle's area, but only the absolute value is useful in this context.
Remember, the determinant captures geometric properties like area, making it quite powerful.

Vertices Coordinates

Vertices are the corner points of the triangle and are given as coordinates in the form \( (x, y) \). Understanding their role is pivotal when it comes to geometry problems like finding area.

• Each vertex contributes its coordinates to form the lines making up the triangle.
• In our example, the vertices are \( P(-1,0), Q(-3,5), R(5,2) \).

These coordinates help in constructing the essential matrix for the determinant calculation, leading directly to the area without using more conventional methods like base-height.
Handling vertex coordinates properly will allow you to translate geometric shapes into numerical forms quickly.

Matrix

A matrix is an ordered array of numbers or elements arranged in rows and columns. In our case, we use a 3x3 matrix to capture the information of the triangle's vertices:

• This matrix helps to coordinate each vertex's location in the space.
• The structure we focus on is: \[ \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \ \end{vmatrix} \]

Each row corresponds to one vertex of the triangle. The `1` is added to accommodate the specific determinant formula needed for area calculation.
Mastering matrices opens doors to solving many spatial and geometric problems efficiently.

Absolute Value

The absolute value in mathematics refers to the non-negative value of a number without regard to its sign. When computing the area of a triangle using the determinant method, we focus on the absolute value:

• The determinant \(D\) might be negative due to vertex orientation, but area is always positive.
• To find the triangle's area, first compute \(|D|\), the absolute value.

In our calculation:
\(D = -34\), so \(|D| = 34\).
The area, being half of this absolute value: \(\frac{1}{2} \times 34 = 17\).
This ensures the area is communicated correctly, irrespective of the direction vertices are listed.