Pythagorean Theorem: Isosceles triangles

Pythagorean Theorem and Isosceles Right Triangles 📐

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it's expressed as $a^2 + b^2 = c^2$, where $c$ is the hypotenuse, and $a$ and $b$ are the other two sides.

An isosceles right triangle (also known as a 45-45-90 triangle) has two sides of equal length. These equal sides are the legs (or cathetus) of the right triangle, and the third side is the hypotenuse.

Applying the Theorem 🧮

Let's denote the length of each of the equal sides as $a$. Since it's an isosceles triangle, $a = b$. We can then rewrite the Pythagorean Theorem as:

$a^2 + a^2 = c^2$

Simplifying, we get:

$2a^2 = c^2$

To find the length of the hypotenuse $c$, we take the square root of both sides:

$c = \sqrt{2a^2} = a\sqrt{2}$

Example Calculation 💡

Suppose we have an isosceles right triangle where each of the equal sides (legs) is 5 units long. We want to find the length of the hypotenuse.

Using the formula $c = a\sqrt{2}$, where $a = 5$:

$c = 5\sqrt{2}$

So, the length of the hypotenuse is $5\sqrt{2}$ units.

Finding a Leg Given the Hypotenuse 🤔

If you know the length of the hypotenuse $c$ and want to find the length of the legs $a$, you can rearrange the formula:

$a = \frac{c}{\sqrt{2}}$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$:

$a = \frac{c\sqrt{2}}{2}$

Code Example 💻

Here's a Python code snippet to calculate the hypotenuse of an isosceles right triangle, given the length of a leg:

import math

def calculate_hypotenuse(leg_length):
    """Calculates the hypotenuse of an isosceles right triangle."""
    return leg_length * math.sqrt(2)

# Example usage:
leg = 5
hypotenuse = calculate_hypotenuse(leg)
print(f"The hypotenuse of an isosceles right triangle with leg {leg} is: {hypotenuse}")

This code snippet uses the math.sqrt() function to calculate the square root of 2 and then multiplies it by the length of the leg to find the hypotenuse.