Explanation
The Pythagorean Theorem describes the relationship between the sides of a right triangle.
It states that:
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this lesson, we prove this visually by comparing two different shapes made from the same triangles.
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 use \(4\) identical right triangles with sides \(a, b, c\).
First Shape
- Arrange the triangles to form two rectangles
- One rectangle is horizontal, the other is vertical
- Together they form a large square with side \((a + b)\)
Inside this square:
- There is a square with side \(a\) → area \(= a^2\)
- There is a square with side \(b\) → area \(= b^2\)
So, the total area inside is:
\(a^2 + b^2\)
Second Shape
- Arrange the same \(4\) triangles differently
- Rotate each triangle by \(90^\circ\)
- This creates a large square with side \((a + b)\)
Inside this square:
- A smaller square is formed
- Each side of this square is \(c\)
So, the area inside is:
\(c^2\)
Conclusion
Both shapes use the same triangles and form the same large square.
So their areas must be equal:
\(a^2 + b^2 = c^2\)
Video Explanation
Practice
Continue Learning
- Pythagorean Formula
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 1
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 2 (Current lesson)
- Pythagoras Theorem (Example)