Pythagoras Theorem

Definition: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Hypotenuse² = Base² + Perpendicular²

In symbolic form, if the sides are a, b (legs) and c (hypotenuse):

c² = a² + b²

Where:

• c = hypotenuse (longest side, opposite the 90° angle)
• a = one leg (base or perpendicular)
• b = the other leg

Important:

• The hypotenuse is always the longest side of a right-angled triangle.
• The theorem does NOT apply to non-right-angled triangles.
• The converse: if c² = a² + b² for a triangle with sides a, b, c, the angle opposite c is 90°.

Proof of Pythagoras Theorem (using similar triangles):

Given: ▵ABC is right-angled at B, with AB = a, BC = b, and AC = c (hypotenuse).

To prove: c² = a² + b²

Construction: Draw BD perpendicular to AC, where D lies on AC.

Proof:

1. In ▵ABC and ▵ADB:
   • ∠A is common
   • ∠ABC = ∠ADB = 90°
   By AA similarity: ▵ABC ~ ▵ADB
   Therefore: AB/AD = AC/AB ⇒ AB² = AD × AC ⇒ a² = AD × c ... (i)
2. In ▵ABC and ▵BDC:
   • ∠C is common
   • ∠ABC = ∠BDC = 90°
   By AA similarity: ▵ABC ~ ▵BDC
   Therefore: BC/DC = AC/BC ⇒ BC² = DC × AC ⇒ b² = DC × c ... (ii)
3. Adding (i) and (ii):
   • a² + b² = AD × c + DC × c
   • a² + b² = c(AD + DC)
   • a² + b² = c × c (since AD + DC = AC = c)
   • a² + b² = c²

Hence proved. ◻

Classifying triangles using the Pythagoras relationship:

1. Right-angled triangle

• If c² = a² + b², the triangle is right-angled at the vertex opposite the longest side c.
• The Pythagoras Theorem applies directly.

2. Acute-angled triangle

• If c² < a² + b² (where c is the longest side), all angles are less than 90°.
• The Pythagoras Theorem does not hold, but the inequality helps classify the triangle.

3. Obtuse-angled triangle

• If c² > a² + b² (where c is the longest side), the angle opposite c is greater than 90°.
• The Pythagoras Theorem does not hold for this triangle type.

4. Pythagorean triplets

• Sets of three positive integers (a, b, c) satisfying a² + b² = c².
• If (a, b, c) is a triplet, then (ka, kb, kc) is also a triplet for any positive integer k.
• Primitive triplets have GCD(a, b, c) = 1.

5. Converse of Pythagoras Theorem

• If a triangle has sides a, b, c with c² = a² + b², then the triangle is right-angled.
• This converse is used to verify whether a given triangle is right-angled.

Applications of Pythagoras Theorem:

• Construction and architecture: Builders use the 3-4-5 rule to verify right angles at corners of buildings, rooms, and foundations.
• Navigation: Pilots and sailors calculate the shortest distance (displacement) between two points after travelling along perpendicular directions.
• Coordinate geometry: The distance formula between two points is a direct application of the theorem.
• Height and distance problems: Finding the height of towers, buildings, and poles when the shadow length and line of sight are known.
• Diagonal calculations: Finding diagonals of rectangles, squares, cuboids, and other geometric shapes.
• Surveying and mapping: Surveyors use the theorem to measure inaccessible distances by creating right triangles.
• Physics: Resolving forces and velocities into perpendicular components uses the Pythagorean relationship.
• Computer graphics: Calculating pixel distances on screen relies on the theorem in the coordinate plane.