Problem 50 - Thomas Calculus

Question

Triangle area Find a concise \(3 \times 3\) determinant formula that gives the area of a triangle in the \(x y\) -plane having vertices \(\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right),\) and \(\left(c_{1}, c_{2}\right)\)

Step-by-Step Solution

Answer

Triangle area from vertices can be found by computing half of the absolute value of a determinant formula based on vertex coordinates.

Step 1: Understand the Triangle Area Formula

The area of a triangle with vertices at \((a_{1}, a_{2}), (b_{1}, b_{2}), (c_{1}, c_{2})\) can be calculated using the formula: \[\text{Area} = \frac{1}{2} \left| a_{1}(b_{2} - c_{2}) + b_{1}(c_{2} - a_{2}) + c_{1}(a_{2} - b_{2}) \right|\]. This formula is derived from the determinant of a 3x3 matrix, which helps in giving the signed area of the triangle based on its vertices' coordinates.

Step 2: Construct the Determinant

To find the determinant, we start with constructing a 3x3 matrix, where each row corresponds to the coordinates of one vertex.\[\begin{array}{ccc}a_{1} & a_{2} & 1 \b_{1} & b_{2} & 1 \c_{1} & c_{2} & 1\end{array}\]

Step 3: Calculate the Determinant

Calculate the determinant of the matrix as follows:\[\det = a_{1}(b_{2}\cdot1 - 1\cdot c_{2}) - a_{2}(b_{1}\cdot1 - c_{1}\cdot1) + 1(b_{1}\cdot c_{2} - c_{1}\cdot b_{2})\]This simplifies to:\[det = a_{1}(b_{2} - c_{2}) - a_{2}(b_{1} - c_{1}) + (b_{1}c_{2} - c_{1}b_{2})\]

Step 4: Triangle Area Formula from Determinant

The area \(A\) of the triangle is half of the absolute value of this determinant:\[A = \frac{1}{2} \left| \det \right|\]Substitute the determinant expression from Step 3:\[A = \frac{1}{2} \left| a_{1}(b_{2} - c_{2}) - a_{2}(b_{1} - c_{1}) + (b_{1}c_{2} - c_{1}b_{2}) \right|\]

Key Concepts

Triangle Area

The area of a triangle is a fundamental concept in geometry that measures the size of a two-dimensional shape. When dealing with vertices located on a coordinate plane, the area can be calculated using specific formulas. One of the well-known methods involves using the coordinates of the points that define the vertices of the triangle. If you have a triangle with vertices

• (a_1, a_2)
• (b_1, b_2)
• (c_1, c_2)

then, the area is calculated by the formula:\[\text{Area} = \frac{1}{2} \left| a_1(b_2 - c_2) + b_1(c_2 - a_2) + c_1(a_2 - b_2) \right|\]This formula is useful because it leverages coordinate geometry and matrix concepts to determine the spatial measurement of the triangle.
The formula is derived from interpreting the triangle as part of a determinant matrix.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate system. The ability to define points in space through coordinates makes solving complex geometric problems more intuitive. In the context of finding a triangle's area, coordinate geometry provides:

• Precise calculations: It's possible to calculate lengths and areas exactly using algebraic formulas.
• Connections between algebra and geometry: Allows the use of algebraic techniques to solve geometric problems.
• Visualization: Helps in visualizing geometric concepts graphically, which aids understanding.

Using coordinate geometry, the vertices of the triangle are easily applied in the formula:\[\text{Area} = \frac{1}{2} \left| a_1(b_2 - c_2) + b_1(c_2 - a_2) + c_1(a_2 - b_2) \right|\]
Applying coordinates directly into this formula helps establish a clear connection between the physical space the points occupy and the algebraic expression of the area.

Matrix Determinant

Matrix determinants are mathematical tools that provide important information about matrices, such as their invertibility and linear transformations associated with areas and volumes. For a triangle on the coordinate plane, using the vertices, we can setup:
\[\begin{array}{ccc}a_{1} & a_{2} & 1 \b_{1} & b_{2} & 1 \c_{1} & c_{2} & 1\end{array}\]Calculating the determinant of this 3x3 matrix helps us derive the formula for the triangle's area. The determinant is found as follows:
\[ ext{det} = a_1(b_2 \cdot 1 - 1 \cdot c_2) - a_2(b_1 \cdot 1 - c_1 \cdot 1) + 1(b_1 \cdot c_2 - c_1 \cdot b_2)\]This simplifies to the triangle area determinant formula and finally converts to the absolute area by dividing by two:
\[A = \frac{1}{2} \left| ext{det} \right|\]By using matrix determinants, this approach highlights the deep interrelationship between algebra and geometry. It also significantly streamlines the process of area calculation in coordinate geometry.