Question

Write and test a Python script that reads in two floating-point side lengths of a right triangle, and outputs the hypotenuse length, perimeter, and area.

Short answer

The Python script calculates hypotenuse, perimeter, and area from two side lengths of a right triangle.

Step-by-step solution

Define the Problem and Inputs

We need to write a Python script that calculates the hypotenuse, perimeter, and area of a right triangle. The inputs will be the lengths of the two legs of the triangle, denoted as "a" and "b", which are floating-point numbers.

Calculate the Hypotenuse

The hypotenuse "c" is found using the Pythagorean theorem, which states: \( c = \sqrt{a^2 + b^2} \). We can use Python's `math.sqrt()` function to calculate this.

Calculate the Perimeter

The perimeter "P" of the right triangle is the sum of all sides: \( P = a + b + c \). By calculating the hypotenuse in the previous step, you can easily compute the perimeter.

Calculate the Area

The area "A" of the right triangle can be found using the formula: \( A = \frac{1}{2} \times a \times b \). This is because the two legs form the base and height of the triangle.

Implement the Script

Start by importing the `math` module, then use input statements to read the side lengths. Calculate the hypotenuse, perimeter, and area using formulas from previous steps, and use `print()` to output these values. ```python import math a = float(input("Enter the length of side a: ")) b = float(input("Enter the length of side b: ")) c = math.sqrt(a**2 + b**2) perimeter = a + b + c area = 0.5 * a * b print(f"Hypotenuse: {c}") print(f"Perimeter: {perimeter}") print(f"Area: {area}") ``` This script will prompt the user for input and then calculate and display the desired values.

Test the Script

Test the script with different pairs of inputs to ensure accuracy. For example, input `a = 3` and `b = 4`. The hypotenuse should be \(5.0\), the perimeter should be \(12.0\), and the area should be \(6.0\). Make sure the output values match these expected results.

Key concepts

Pythagorean theorem

The Pythagorean theorem is one of the cornerstones of geometry. It applies specifically to right triangles, which have one angle measuring 90 degrees. According to the theorem, the square of the length of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This can be represented by the formula: \[ c^2 = a^2 + b^2 \]where:

• \( c \) is the length of the hypotenuse
• \( a \) and \( b \) are the lengths of the other two sides

To find the hypotenuse \( c \), we rearrange the formula and take the square root:\[ c = \sqrt{a^2 + b^2} \]Python provides a handy function called `math.sqrt()` that we use in programming to compute this square root efficiently.
This theorem is essential not only in geometry but also in many practical fields such as construction, navigation, and physics.

right triangle calculations

Calculating dimensions of a right triangle involves understanding the relationships between its sides and angles. In our Python script, we focus on calculating three key values: the hypotenuse, perimeter, and area. Knowing these will help in various applications such as navigation and engineering.

• Hypotenuse:
  By using the Pythagorean theorem, the hypotenuse is easily calculated once we have the lengths of the other two sides.
• Perimeter:
  The total length around the triangle can be determined by adding all its side lengths:\[ P = a + b + c \]where \( c \) is the hypotenuse calculated earlier.
  This is straightforward once you've calculated the hypotenuse.
• Area:
  The formula for the area of a right triangle considers the two other sides (legs) as the base and height and is given by:\[ A = \frac{1}{2} \times a \times b \]This simplicity in computation makes it quite user-friendly.
  Areas are particularly useful in scenarios involving land measurement, architectural layouts, etc.

floating-point arithmetic

When using programming languages like Python, we often deal with floating-point numbers. These are numbers that have a decimal point and are essential in accurately representing non-integer values.

• Precision:
  Floating-point numbers can be more precise than integers, crucial for measurements that require greater accuracy.
  However, small rounding errors may still occur due to how these numbers are stored in binary format.
• Input and Output:
  In programming scripts, especially when dealing with user inputs, it is common to read inputs as float type to accommodate decimals. This flexibility allows us to handle complex calculations, such as those in geometry where measurements are rarely whole numbers.
• Computational Implications:
  When performing arithmetic operations on floating-point numbers, remember that operations like addition or multiplication might result in minor precision errors.
  It is essential to keep this in mind for applications needing high mathematical precision, such as scientific computations.

By understanding floating-point arithmetic, we ensure that our calculations are as accurate as possible and accessible for every potential real-world scenario.