Question

Given \(a > c > 0\) and \(b > d > 0\) define \(\Delta A B C\) to be the triangle with vertices \(A=(0,0), B=(a, b)\) and \(C=(c, d) .\) Prove that \(\triangle A B C\) has area \(\frac{1}{2}(b c-a d) .\) Use this to find a point \(C\) such that \(\triangle A B C\) will be fundamental, for \(B=(5,3)\)

Short answer

To make \(\triangle ABC\) fundamental with \(B = (5,3)\), a point \(C = (1,1)\) can be used.

Step-by-step solution

Understanding the Area Formula

The area of a triangle given vertex coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) can be calculated using the determinant method: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|. \] Applying this to our vertices, we have \(A = (0,0), B = (a, b), C = (c, d)\). Substituting these values into the formula, it simplifies to \[ \text{Area} = \frac{1}{2}\left| 0(b-d) + a(d-0) + c(0-b) \right| = \frac{1}{2}\left| ad - bc \right|. \] Thus, the area is \(\frac{1}{2}(ad - bc)\).

Given Area Condition for Reference Triangle

A triangle is said to be fundamental if its area is \(1\). With \(B = (5,3)\) already given, we need to find a point \(C = (c,d)\) such that the area of triangle \(\Delta A B C\) is \(1\). From our earlier expression for the area, we require \[ \frac{1}{2}(5d - 3c) = 1. \] Simplifying, this gives us the equation \(5d - 3c = 2\).

Solve for Possible Coordinates of C

To find possible coordinates for \(C = (c,d)\), we need to satisfy the equation \(5d - 3c = 2\). We can choose integer values for \(d\) and solve for \(c\), or vice versa. For example, setting \(d = 1\), we have: \[ 5(1) - 3c = 2 \] \[ 5 - 3c = 2 \] \[ 3c = 3 \] \[ c = 1. \] Thus, \(C = (1,1)\) is one such solution. We could also experiment with other values to find additional solutions.

Key concepts

Vertex Coordinates

When dealing with triangles in the coordinate plane, it's essential to understand how vertices are defined. Each vertex of a triangle is specified by its coordinates. For instance, if a triangle has vertices labeled as points \(A, B,\) and \(C\), their coordinates are written as \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) respectively.
In our specific example, Vertex \(A\) is at \((0,0)\), making it highly convenient since it's at the origin. Vertex \(B\) is at \((a, b)\), and Vertex \(C\) is at \((c, d)\). The position of each vertex directly influences the area of the triangle through geometric calculations like those used in the determinant method.
When calculating areas or other properties, it’s critical first to identify these vertex coordinates correctly. This ensures that all subsequent calculations, such as area, perimeter, and other geometric properties, will be accurate.

Determinant Method

The determinant method is a mathematical technique used to calculate the area of a triangle when its vertices are known. Using a determinant is efficient because it allows you to plug the coordinates of the triangle's vertices into a formula, yielding the area quickly and accurately.
The general formula is:

• \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|. \]

In our case with vertices \((0,0), (a, b), (c, d)\), this simplifies to \[ \frac{1}{2}\left| ad - bc \right|. \] This formula is derived by rearranging the coordinates in a pattern that resembles a determinant of a matrix. It takes into account both the horizontal and vertical components of the triangle relative to the coordinate plane.
Utilizing this method helps not only in calculating areas but also in understanding the geometric relationships between the vertices. It's a fundamental tool in coordinate geometry, enabling more complex spatial analyses.

Fundamental Triangle

A triangle is considered a fundamental triangle in geometry if it has a particular, usually simplified, characteristic. In plane geometry, a fundamental triangle is often one whose area is equal to a specified value, which in many exercises is \(1\).
In our problem, the goal is to find a point \(C\) such that the area of \(\Delta ABC\) is \(1\). This involves ensuring the formula \( \frac{1}{2}(ad - bc) = 1 \) holds true. Solving this will provide specific coordinates \((c, d)\) for Vertex \(C\) that make the triangle fundamental.
Finding such a triangle usually involves some trial and error, or strategic selection of integers to solve the constraint equation efficiently. This ensures the triangle remains fundamental, and can be a useful strategy in various optimization problems or in constructing particular geometric configurations.

Geometric Proof

A geometric proof involves logical reasoning to demonstrate the truth of a geometric statement or property. In this context, using mathematical equations to derive and verify properties, such as the area formula for a triangle, epitomizes a geometric proof.
Proofs often start with given information—in our case, vertex coordinates—and through a series of logical steps and formula applications, a truth is established. Here, we've used the determinant method as a key step in showing the area of a triangle given three points. Each element of the proof must be justified, either through deductive logic or mathematical identity.
Beyond just the numbers, a geometric proof provides deeper insight into how geometric shapes work and interact. It’s a way to confirm the results of computations and ensure that understanding remains aligned with theoretical principles, such as those governing areas and their calculation.