Problem 43 - 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. $$P(1,2), Q(4,3), R(3,5)$$

Step-by-Step Solution

Answer

The area of triangle PQR is 3.5 square units.

Step 1: Write the coordinates

Identify the coordinates of the vertices of the triangle. These are given as: \( P(1, 2), Q(4, 3), R(3, 5) \).

Step 2: Set up the determinant

To find the area of the triangle, we use the determinant formula. The determinant \( D \) is given as \( \frac{1}{2} \cdot \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| \). For the vertices \( P(1, 2), Q(4, 3), R(3, 5) \), this becomes: \[ D = \frac{1}{2} \cdot \left| \begin{array}{ccc} 1 & 2 & 1 \ 4 & 3 & 1 \ 3 & 5 & 1 \end{array} \right| \].

Step 3: Calculate the determinant

Calculate the determinant of the matrix: \[ \left| \begin{array}{ccc} 1 & 2 & 1 \ 4 & 3 & 1 \ 3 & 5 & 1 \end{array} \right| = 1(3 \cdot 1 - 1 \cdot 5) - 2(4 \cdot 1 - 1 \cdot 3) + 1(4 \cdot 5 - 3 \cdot 3). \] This simplifies to: \[ 1(3 - 5) - 2(4 - 3) + 1(20 - 9) \] = \(-2 - 2 + 11\) = \(7\).

Step 4: Compute the area

Substitute the value of the determinant back into the formula for the area: \[ \text{Area} = \frac{1}{2} \cdot |7| = \frac{1}{2} \cdot 7 = 3.5. \]

Step 5: Conclude the result

The area of the triangle with vertices \( P(1, 2), Q(4, 3), R(3, 5) \) is 3.5 square units.

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle using its vertices in a coordinate plane is a classic geometry problem. The formula involves determining the absolute value of a determinant. For a triangle with vertices at coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the area \(A\) is found by using:

\[ A = \frac{1}{2} \left| D \right| \]
where \(D\) is the determinant of a 3x3 matrix.

The simplicity of this method lies in its reliance on matrix algebra, allowing for quick computation without needing traditional methods like Heron's formula.

Steps include:

• Identifying the coordinates of each vertex.
• Setting up the determinant using these coordinates.
• Calculating the determinant value.
• Taking half the absolute value of this determinant to find the area.

This technique is particularly useful in coordinate geometry because it leverages known algebraic methods to find geometric properties.

Matrix Determinants

Matrix determinants are a fundamental concept in linear algebra, often used to solve systems of equations, but they also have geometric applications. A determinant is a scalar value that provides important information about a matrix. For a 3x3 matrix with elements \( a_{ij} \), the determinant can be calculated as:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]
This formula considers different permutations of multiplication and subtraction of elements in the matrix, encapsulating its geometric properties.

In our triangle area problem, the matrix includes an extra column of 1s to account for the plane transformation, which simplifies calculations and aligns with geometric interpretations of position and area.

The determinant thus connects algebra and geometry, transforming coordinate points into measurable quantities, like areas and volumes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves representing geometric shapes in a numerical system. It merges algebra and geometry into a cohesive framework, enabling mathematical analysis and calculations. By placing triangles within the coordinate plane, we can use algebraic techniques to explore geometric properties.

Some key points in coordinate geometry include:

• Using coordinates to describe points and shapes.
• Applying algebraic equations to find distances, slopes, and shapes.
• Translating visual shapes into mathematical models for easier manipulation.

In the context of our exercise, coordinate geometry allows us to find the area of a triangle through matrix determinants. This interplay highlights how coordinate geometry simplifies complex geometric problems by assigning numerical values to points and using those values in algebraic constructs.