Pythagoras Theorem

The Pythagoras Theorem is a fundamental rule in geometry that applies only to right-angled triangles. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Symbolically:

\( a^2 + b^2 = c^2 \)

       Right Triangle
            |
        b   |    c (hypotenuse)
            |____
              a

In a right-angled triangle:

• The side opposite the right angle is the hypotenuse (the longest side).
• The other two sides are called the legs or perpendicular and base.

Let:

• \(c\) = hypotenuse
• \(a\) = one leg
• \(b\) = the other leg

Then the theorem is:

\( a^2 + b^2 = c^2 \)

The theorem comes from the relationship between the areas of squares constructed on each side of a right triangle. The large square on the hypotenuse has exactly the same area as the sum of the squares on the two smaller sides.

This geometric connection ensures the formula always holds true.

     a² + b² = c²

    [Square on side a]
    #######
    #######

    [Square on side b]
    ********
    ********

    [Square on side c]
    ###################
    ###################

The theorem can be used to:

• Find the length of the hypotenuse
• Find the length of a missing leg
• Check whether a triangle is a right triangle

Examples:

If two sides are known, you can find the third:

• If \( a = 6 \), \( b = 8 \), then \( c = \sqrt{6^2 + 8^2} = 10 \)
• If \( c = 13 \), \( b = 5 \), then \( a = \sqrt{13^2 - 5^2} = 12 \)

The converse also holds:

If for a triangle, \( a^2 + b^2 = c^2 \), then the triangle must be a right-angled triangle.

This is used in many geometric proofs and constructions.

This theorem is widely used in:

• Construction and architecture
• Navigation and mapping
• Physics (vectors and forces)
• Computer graphics
• Distance formulas in coordinate geometry
• Finding heights and distances using ladders, poles, and shadows

It is one of the most important and frequently used theorems in mathematics.