The proof of Pythagorean Theorem in mathematics is very important.

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

States that in a right triangle that, the square of a (a

In short it is written as: a

Let QR = a, RP = b and PQ = c. Now, draw a square WXYZ of side (b + c). Take points E, F, G, H on sides WX, XY, YZ and ZW respectively such that WE = XF = YG = ZH = b.

Then, we will get 4 right-angled triangle, hypotenuse of each of
them is ‘a’: remaining sides of each of them are band c. Remaining part of the
figure is the

Now, we are sure that square WXYZ = square EFGH + 4 ∆ GYF

or, (b + c)

or, b

or, b

Proof of Pythagorean Theorem using Algebra:

**Construction:** Draw YO ⊥ XZ

**Proof:** In ∆XOY and ∆XYZ, we have,

∠X = ∠X → common

∠XOY = ∠XYZ → each equal to 90°

Therefore, ∆ XOY ~ ∆ XYZ → by AA-similarity

⇒ XO/XY = XY/XZ

⇒ XO × XZ = XYIn ∆YOZ and ∆XYZ, we have,

∠Z = ∠Z → common

∠YOZ = ∠XYZ → each equal to 90°

Therefore, ∆ YOZ ~ ∆ XYZ → by AA-similarity

⇒ OZ/YZ = YZ/XZ

⇒ OZ × XZ = YZFrom (i) and (ii) we get,

XO × XZ + OZ × XZ = (XY

⇒ (XO + OZ) × XZ = (XY

⇒ XZ × XZ = (XY

⇒ XZ

Conditions for the Congruence of Triangles

Right Angle Hypotenuse Side congruence

Converse of Pythagorean Theorem

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