Pythagoras Theorem

What is Pythagoras Theorem?

Pythagoras Theorem is one of the most fundamental results in geometry. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically: If $\triangle ABC$ is right-angled at $C$, then $AB^2 = AC^2 + BC^2$, or equivalently $c^2 = a^2 + b^2$.

Understanding the Terms

Hypotenuse: The longest side of a right triangle, always opposite the 90° angle.

Legs (Base and Perpendicular): The two sides that form the right angle. Either leg can be called 'base' or 'perpendicular' — the names are interchangeable.

Important: Pythagoras theorem ONLY applies to right-angled triangles. For other triangles, you need the cosine rule (studied in Class 11).

Proof of Pythagoras Theorem (CBSE Syllabus)

The proof prescribed by CBSE uses the concept of similar triangles:

Given: $\triangle ABC$ right-angled at $B$.

To prove: $AC^2 = AB^2 + BC^2$.

Construction: Draw $BD \perp AC$.

Proof: In $\triangle ADB$ and $\triangle ABC$: $\angle A$ is common, $\angle ADB = \angle ABC = 90°$. By AA similarity, $\triangle ADB \sim \triangle ABC$.

So $\frac{AD}{AB} = \frac{AB}{AC}$, giving $AD \cdot AC = AB^2$ ... (i)

Similarly, $\triangle BDC \sim \triangle ABC$, giving $CD \cdot AC = BC^2$ ... (ii)

Adding (i) and (ii): $(AD + CD) \cdot AC = AB^2 + BC^2$, i.e., $AC \cdot AC = AB^2 + BC^2$, i.e., $AC^2 = AB^2 + BC^2$. Proved.