Pythagoras’ theorem — Proof using Similarity of Triangles
There are various methods to prove the Pythagoras’ theorem. This method makes use of similarity of triangles.
Statement of Pythagoras’ theorem:
In any right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of other two sides.
Construction:
Draw segment AD perpendicular to side BC.
Notation:
Let the length of AB be c, length of BC be a and length of AC be b.
Let the length of BD be x and the length of DC be y.
So, a = x + y .
Let m∠C =α and m∠B =β .
By chasing the other angles in right angled triangles ∆ADC and ∆ABD, we get m∠CAD = β and m∠BAD = α respectively.
Restatement:
In ∆ABC, if m∠A = 90°, then b² + c² = a².
Proof:
Step 1:
Prove that h²= xy .
∆ADB~∆CDA~∆CAB ………..using angle-angle-angle (A-A-A) similarity of triangles.(Result 1)
Using Result 1,
AD / DB = CD / DA
h / x = y / h
∴ h²= xy
Step 2:
Prove that b²= ay .
Using Result 1, we can say that
CA / CD = CB / CA
b / y = a / b
∴ b²= ay
Step 3:
Similarly we can prove that c²= ax .
Step 4:
Consider left hand side (L.H.S) of the equation i.e. b² + c² .
b² + c² = ay + ax = a(x+y) = a² .
∴ b² + c² = a²