Problem 61 - Precalculus

Question

Apply determinants to find the area of a triangle with vertices, \((3,2),(5,2),\) and \((3,-4) .\) Check your answer by plotting these vertices in a Cartesian plane and using the formula for area of a right triangle.

Step-by-Step Solution

Answer

The area of the triangle is 6 square units.

Step 1: Arrange Vertices for Determinant Calculation

To find the area of the triangle using a determinant, arrange the vertices \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) into a matrix:\[ \begin{vmatrix} 3 & 2 & 1 \5 & 2 & 1 \3 & -4 & 1 \end{vmatrix}\]

Step 2: Calculate the Determinant

Use the formula for the determinant of a 3x3 matrix to calculate:\[ \text{Determinant} = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\]Replacing the points:\[ = 3(2 - (-4)) + 5((-4) - 2) + 3(2 - 2)\]Simplify:\[ = 3(6) + 5(-6) + 0\]\[ = 18 - 30 + 0 = -12\]

Step 3: Find Absolute Value and Compute Area

Take the absolute value of the determinant and divide by 2 to find the area:\[ \text{Area} = \frac{|\text{Determinant}|}{2} = \frac{|-12|}{2} = \frac{12}{2} = 6\]

Step 4: Validation with Right Triangle Area Formula

Plot the points on the Cartesian plane. Notice that the triangle forms a right triangle with base running from \(3,2\) to \(5,2\) and height from \(3,2\) to \(3,-4\):- Base length = 2- Height length = 6Area = \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 6 = 6\).

Key Concepts

Area of a Triangle

Finding the area of a triangle can be intriguing and insightful, especially when using determinants. The determinant method is a straightforward mathematical technique to calculate the area when you know the vertices of the triangle. If you have a triangle with vertices

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

you can arrange these into a 3x3 matrix to determine the area. The formula for calculating the determinant, which helps to establish the area, is:\[ ext{Area} = \frac{1}{2} \times |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|\]Here, the absolute value of the determinant ensures that the area is always positive, as it represents a geometric space. It's fascinating how this method leverages algebra to solve a geometry problem. This mathematical connection shows how different branches of math often overlap.

Cartesian Plane

The Cartesian plane is a fundamental concept in geometry and algebra that helps us visually interpret mathematical ideas. It consists of two perpendicular lines:

• the x-axis (horizontal)
• the y-axis (vertical)

These axes intersect at a point called the origin, designated as \((0, 0)\).Every point on the plane can be specified by a pair of coordinates \((x, y)\), which tells you how far along the x and y axes the point is.
Using the Cartesian plane, you can easily plot the vertices of a triangle and visualize its dimensions. This visualization is crucial when checking your work or ensuring the integrity of your calculations. By plotting \((3, 2)\), \((5, 2)\), and \((3, -4)\) on this plane, you can see the triangle's structure, making it easier to apply geometry formulas. Additionally, visualizing can sometimes reveal properties, such as whether a triangle is a right triangle, adding further validation to your solution.

Right Triangle

A right triangle is a type of triangle where one of its angles measures exactly \(90^\circ\). The presence of this right angle simplifies many calculations, including that of finding the area. In a right triangle, the legs form the base and the height, typically standing perpendicular to each other.
The formula for the area of a right triangle is:\[ ext{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]This simple expression works because the base and height are always at right angles, allowing for straightforward multiplication.
In our example, plotting the points \((3, 2)\), \((5, 2)\), and \((3, -4)\) revealed a right triangle, where the base is the horizontal segment from \((3,2)\) to \((5,2)\), with length 2, and the height is the vertical segment from \((3,2)\) to \((3, -4)\), measuring 6.This confirms the calculated area of the triangle using the determinant, showcasing the harmony between algebraic determinants and geometric visualization.