Proof of Pythagoras Theorem

Pythagoras Theorem:

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

If triangle ABC is right-angled at B, then:

AC² = AB² + BC²

where:

• AC = hypotenuse (the side opposite the right angle — always the longest side)
• AB and BC = the other two sides (legs)

Converse of Pythagoras Theorem:

If in a triangle, the square of one side equals the sum of squares of the other two sides, then the angle opposite to that side is a right angle.

If AC² = AB² + BC², then angle B = 90°.

Key triangle relationships from the proof:

• When the altitude BD is drawn from the right angle B to hypotenuse AC, three similar triangles are formed: Triangle ABC ~ Triangle ADB ~ Triangle BDC.
• Each leg of the original triangle is the geometric mean of the hypotenuse and the adjacent segment: AB^2 = AD x AC and BC^2 = DC x AC.
• The altitude to the hypotenuse is the geometric mean of the two segments: BD^2 = AD x DC.
• The altitude to the hypotenuse satisfies: 1/BD^2 = 1/AB^2 + 1/BC^2.

Proof of Pythagoras Theorem (Using Similar Triangles — NCERT Method):

Given: Triangle ABC, right-angled at B (angle B = 90°).

To prove: AC² = AB² + BC².

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

Proof:

1. In triangle ADB and triangle ABC:
   • Angle ADB = angle ABC = 90°
   • Angle A = angle A (common)
   • By AA similarity: triangle ADB is similar to triangle ABC.
2. Therefore: AD/AB = AB/AC (corresponding sides of similar triangles).
   Cross-multiply: AB² = AD × AC ... (i)
3. In triangle BDC and triangle ABC:
   • Angle BDC = angle ABC = 90°
   • Angle C = angle C (common)
   • By AA similarity: triangle BDC is similar to triangle ABC.
4. Therefore: DC/BC = BC/AC (corresponding sides of similar triangles).
   Cross-multiply: BC² = DC × AC ... (ii)
5. Add equations (i) and (ii):
   AB² + BC² = AD × AC + DC × AC
   = AC(AD + DC)
   = AC × AC (since AD + DC = AC)
   = AC²

Therefore: AC² = AB² + BC².

QED.

Key insight: The altitude from the right angle vertex to the hypotenuse creates two smaller triangles, each similar to the original triangle and to each other. This triple similarity is the foundation of the proof.

Types of Pythagoras Theorem problems:

Type 1: Finding the hypotenuse

• Given two legs, find the hypotenuse.
• Formula: c = √(a² + b²)
• Example: Legs = 6, 8 → Hypotenuse = √(36 + 64) = √100 = 10

Type 2: Finding a leg

• Given the hypotenuse and one leg, find the other leg.
• Formula: a = √(c² − b²)
• Example: Hypotenuse = 13, one leg = 5 → Other leg = √(169 − 25) = √144 = 12

Type 3: Checking for right angle (Converse)

• Given three sides, check if c² = a² + b² (where c is the longest side).
• If yes → right-angled triangle.
• If c² > a² + b² → obtuse triangle.
• If c² < a² + b² → acute triangle.

Type 4: Proof-based problems

• Prove geometric results using the Pythagoras Theorem.
• Example: Prove that in an isosceles right triangle with legs a, the hypotenuse = a√2.
• Example: Prove that the altitude from the right angle vertex to the hypotenuse gives AB² = AD × AC.

Type 5: Word problems

• Ladder against a wall — the ladder is the hypotenuse, the wall is the vertical leg, the ground distance is the horizontal leg.
• Distance between two points — bird on a pole, ships at sea, planes in the sky.
• Diagonal of a rectangle — the diagonal forms the hypotenuse with length and breadth as legs.

Type 6: In other shapes

• Rectangle: Diagonal = √(l² + b²)
• Square: Diagonal = a√2
• Rhombus: Side = √[(d₁/2)² + (d₂/2)²] (diagonals bisect each other at right angles)
• Equilateral triangle: Height = (√3/2) × side
• Coordinate geometry: Distance formula is derived from the Pythagoras Theorem.

Method 1: Direct application

• Identify the right angle and the hypotenuse.
• Apply c² = a² + b² directly.

Method 2: Converse application

• Given three sides, check if the square of the longest side equals the sum of squares of the other two.

Method 3: In compound figures

• Break the figure into right triangles.
• Apply the theorem in each triangle.

Tips:

• Always identify the hypotenuse first — it is the side opposite the right angle and is always the longest.
• The theorem applies ONLY to right-angled triangles.
• Pythagorean triplets (3-4-5, 5-12-13, 8-15-17) allow quick recognition of right triangles.
• When working with surds, leave the answer in surd form unless asked to approximate.

Presentation tips for board exams:

• Draw a neat figure with angle B clearly marked as 90 degrees and BD perpendicular to AC.
• Write the Given and To Prove statements before starting the proof.
• Clearly state the construction: "Draw BD perpendicular to AC."
• When proving similarity, explicitly mention both angle pairs (common angle + right angle).
• Write the proportionality statement from each similarity clearly before cross-multiplying.
• In the final step, clearly show that AD + DC = AC.
• End with "Hence Proved" or "QED".
• Total proof should take about 1 page of writing.

Common exam variations:

• Prove Pythagoras Theorem (full proof) — 5 marks
• Using the proof, find segments of the hypotenuse — 3-4 marks
• Prove the geometric mean relation BD^2 = AD x DC — 3 marks
• Prove 1/BD^2 = 1/AB^2 + 1/BC^2 using the altitude — 4 marks

Applications of the Pythagoras Theorem:

• Construction — builders use the 3-4-5 rule to ensure corners are exactly 90°. Lay out 3 m on one side, 4 m on the other; if the diagonal is 5 m, the corner is right-angled.
• Navigation — calculating straight-line (as-the-crow-flies) distance when north-south and east-west displacements are known. Ships and aircraft use this constantly.
• Architecture — calculating roof heights (the slope forms the hypotenuse), ramp lengths for wheelchair access, staircase dimensions, and diagonal bracing for structural support.
• Coordinate geometry — the distance formula d = √[(x₂−x₁)² + (y₂−y₁)²] is a direct application of the Pythagoras Theorem in the Cartesian plane.
• Physics — resolving vectors into perpendicular components (force, velocity, displacement). The magnitude of the resultant of two perpendicular vectors is found using c = √(a² + b²).
• Computer graphics — pixel distance calculations, collision detection between objects, and determining whether a point lies inside a circle (compare distance to radius).
• Surveying — measuring heights of buildings, mountains, and towers indirectly by forming right triangles with known horizontal distances.
• Everyday life — TV and monitor sizes are measured as diagonals. A "55-inch TV" has a 55-inch diagonal, and the actual width and height are found using the Pythagoras Theorem with the aspect ratio.

Historical significance:

• The theorem was known to Babylonians around 1800 BCE (clay tablet Plimpton 322 lists Pythagorean triplets).
• Indian mathematicians (Sulba Sutras, ~800 BCE) used the theorem for constructing fire altars with precise right angles.
• Pythagoras of Samos (~570–495 BCE) is traditionally credited with the first formal proof.
• Over 400 distinct proofs of the theorem exist, including proofs by US President James Garfield and Leonardo da Vinci.

Further applications of the altitude-on-hypotenuse construction:

• Spiral of Theodorus: A geometric construction that uses repeated right triangles to represent square roots of successive natural numbers. Each new triangle shares the hypotenuse of the previous one as a leg, creating a spiral pattern.
• Golden ratio: The Pythagoras Theorem is used in deriving the golden ratio through the diagonal of a regular pentagon, which involves right triangles with specific proportions.
• Three-dimensional geometry: The 3D distance formula d = sqrt(dx^2 + dy^2 + dz^2) is derived by applying Pythagoras Theorem twice — once in the horizontal plane and once vertically.