Explanation
The Pythagorean Theorem explains the relationship between the sides of a right triangle.
A right triangle has one angle equal to \(90^\circ\).
The theorem states that:
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
But instead of just using the formula, here we will prove why it works using areas.
Formula / Rule
\(a^2 + b^2 = c^2\)
Where:
- \(c\) is the hypotenuse (longest side)
- \(a\) and \(b\) are the other two sides
Example (Visual Idea of the Proof)
We take \(4\) identical right triangles with sides \(a, b, c\).
Step 1
Arrange the \(4\) triangles to form a large square
- Each side of the large square is \((a + b)\)
Step 2
Inside the large square, a small square is formed
- Each side of the small square is \(c\)
Step 3 – Calculate the area in two ways
Method 1: Area of the large square
\begin{align}\text{Area } &= (a + b)^2\\
&= a^2 + 2ab + b^2\end{align}
Method 2: Area using shapes inside
\begin{align}\text{Area } &= \text{ area of }4 \text{ triangles } + \text{ area of small square }\\
&= 4 × (\frac12 \times a \times b) + c^2\\
&= 2ab + c^2\end{align}
Step 4 – Compare both areas
\(a^2 + 2ab + b^2 = 2ab + c^2\)
Cancel \(2ab\) from both sides:
\(a^2 + b^2 = c^2\)
This proves the Pythagorean Theorem.
Video Explanation
Practice
Continue Learning
- Pythagorean Formula
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 1 (Current lesson)
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 2
- Pythagoras Theorem (Example)