Question

Compute the area of an arbitrary triangle. An arbitrary triangle can be described by the coordinates of its three vertices: \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\), numbered in a counterclockwise direction. The area of the triangle is given by the formula $$ A=\frac{1}{2}\left|x_{2} y_{3}-x_{3} y_{2}-x_{1} y_{3}+x_{3} y_{1}+x_{1} y_{2}-x_{2} y_{1}\right| $$ Write a function area(vertices) that returns the area of a triangle whose vertices are specified by the argument vertices, which is a nested list of the vertex coordinates. For example, vertices can be \([[0,0],[1,0],[0,2]]\) if the three corners of the triangle have coordinates \((0,0),(1,0)\), and \((0,2)\). Test the area function on a triangle with known area. Name of program file: area_triangle.py.

Short answer

Use the formula in a function to compute the area of the triangle.

Step-by-step solution

Understand the Formula

The formula for the area of a triangle given its vertices is \( A = \frac{1}{2} \left| x_2 y_3 - x_3 y_2 - x_1 y_3 + x_3 y_1 + x_1 y_2 - x_2 y_1 \right| \). This uses the coordinates of the triangle's vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) and calculates the absolute value of the expression inside the brackets to ensure a positive area, then divides by 2.

Define the Input Format

The input to the function is a list called `vertices`, which contains three sublists. Each sublist represents a vertex of the triangle in the format \([x, y]\). For example, vertices can be \([[0,0],[1,0],[0,2]]\).

Implement the Function

Define a function `area(vertices)` in a Python script named `area_triangle.py`. Use the given formula to compute the area of the triangle by extracting coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) from the `vertices` list. The area is computed as follows:```pythondef area(vertices): x1, y1 = vertices[0] x2, y2 = vertices[1] x3, y3 = vertices[2] return 0.5 * abs((x2 * y3 - x3 * y2) - (x1 * y3 - x3 * y1) + (x1 * y2 - x2 * y1))```

Test the Function

Create a test case to verify that the function works correctly. Use the coordinates \( (0,0), (1,0), (0,2)\) which forms a right triangle and should have an area of 1. Execute the function with the given test inputs:```pythonvertices = [[0, 0], [1, 0], [0, 2]]print(area(vertices)) # Expected output: 1.0```

Key concepts

Triangle Area Calculation

Calculating the area of a triangle using its vertex coordinates is an essential concept in geometry. The formula for finding the area of a triangle from vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) involves a combination of coordinates. This method takes a computational approach which is efficient and works with any arbitrary triangle. The formula is as follows:\[A = \frac{1}{2} \left| x_2y_3 - x_3y_2 - x_1y_3 + x_3y_1 + x_1y_2 - x_2y_1 \right|\]The expression inside the absolute value brackets computes a mathematical determinant using the coordinates. This determinant gives the parallelogram formed by doubling our triangle, hence the division by 2.This formula is particularly useful because it doesn't require side lengths; only the vertex coordinates are needed. Calculating it manually can be cumbersome, which is why writing a function in Python is such a time saver.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use coordinate systems to define and analyze geometric figures. In the problem of calculating a triangle's area, we are precisely using this method by working with coordinate points as inputs. These points form vertices of the triangle.

• A vertex is represented as a point in space: \((x, y)\) in 2D geometry.
• For a triangle, three such vertices uniquely outline the entire shape.
• Using these coordinates, various properties of the triangle such as length of sides, angles, and area can be calculated.

Coordinate geometry serves an integral role in making these mathematical calculations straightforward by using algebraic formulas. The transformation of visual geometric shapes into numerical operations is what long has made it a valuable tool in fields like engineering, architecture, and computer graphics.

Programming Functions

Understanding and implementing functions is a core aspect of programming. A function in programming is a block of reusable code designed to perform a specific task. Here, we are tasked to create a function called `area()` in Python that calculates the area of a triangle given its vertices. Functions offer several benefits:

• They improve code readability by allowing complex operations to be encapsulated in a single action.
• Functions facilitate code reusability, enabling us to apply the same logic with different inputs without rewriting the logic.
• Using functions also allows for easier debugging and testing since each function can be tested independently.

For this task: - The function `area(vertices)` takes one parameter `vertices`, a list of three sublists, each containing two elements - the x and y coordinates. - Inside the function, the coordinates are extracted, and the area is computed using the given formula. This modular approach demonstrates efficient coding practices.

Python Scripting

Python scripting is about writing scripts that automate tasks. In this instance, we write a script to calculate the area of a triangle based on user-provided coordinates. Python is particularly suited for such tasks because of its simple syntax and powerful libraries. Let's break down the process of our script `area_triangle.py` which performs this calculation:

• The script starts by defining a function called `area()`.
• Inside this function, coordinates of the vertices are taken from the `vertices` parameter.
• These coordinates are used in the formula to calculate and return the area of the triangle.
• We then test the function with known values to ensure our script works correctly.

By scripting in Python, we can quickly automate repetitive calculations and ensure consistency and accuracy across different tasks. Additionally, scripts can be easily shared or modified to tackle a wide variety of similar problems.