What is the definition of pythagorean theorem?

Answer 1

It doesn't prove anything.

A "theorem" is a statement that tells how things work.

The Pythagorean theorem tells how the relationship between the lengths of the sides of a
right triangle works.

Sounds pretty dry, I know; but it's fantastically useful. Every branch of science, engineering,
carpentry, Plumbing, and real estate uses it every day.

Answer 2

Consider the diagram:

. D. . . . . P. . . . . . . . . E.

. . -------+------------. .

. . | . . . . . . . . . . . . . . |. .

. . | . . . . . . . . . . . . . . |. .

. . | . . . . . . . . . . . . . .+ Q

. . | . . . . . . . . . . . . . . |. .

S + . . . . . . . . . . . . . . |. .

. . | . . . . . . . . . . . . . . |. .

. . | . . . . . . . . . . . . . . |. .

. . ------------+-------. .

.G. . . . . . . . . .R . . . . .F.

where:

1. DEFG is a square of side length (a+b)
2. Lengths DP = EQ = FR = GS = a
3. Lengths PE = QF = RG = SD = b
4. There are four further lines (but for clarity of diagram with the drawing ability available, they are not shown); join:
   1. PQ to make right triangle PEQ
   2. QR to make right triangle QFR
   3. RS to make right triangle RGS
   4. SP to make right triangle SDP
5. Let length PQ = c

The four lines in (4) above make quadrilateral PQRS.

Consider the four triangles PEQ, QFR, RGS, SDP.

1. Corresponding sides PE, QF, RG, SD are all equal (to length b)
2. Corresponding sides EQ, FR, GS, DP are all equal (to length a)
3. The enclosed angles ∠PEQ, ∠QFR, ∠RGS, ∠SDP are all equal (to a right angle, 90o)

thus the four triangles are congruent (Side-Angle-Side); so:

PQ = QR = RS = SP = c

Now consider one of the sides DPE, say, of the square DEFG:

At P three angles meet: ∠DPS, ∠SPQ, ∠QPE, and since DPE is a straight line:

∠DPS + ∠SPQ + ∠QPE = 180o

Since triangles SDP and PEQ are congruent, ∠QPE = ∠PSD

As SDP is a right triangle with the right angle at D, the angles ∠DPS and ∠PSD are complementary, that is sum to 90o.

thus:

∠DPS + ∠SPQ + ∠QPE = 180o

⇒ ∠DPS + ∠SPQ + ∠PSD = 180o

⇒ ∠SPQ = 180o - (∠DPS + ∠PSD)

= 180o - 90o

= 90o

The same consideration can be made for all the sides of square DEFG, showing all angles are right angles.

Thus since PQRS has equal sides (length c) and equal angles (90o) it is a square.

Now consider the area of square DEFG:

It can be calculated directly:

area = (a+b) x (a+b)

= a2 + b2 + 2ab

or it can be calculated by summing the areas of the square PQRS and the four congruent triangles:

area = c x c + 4 x (1/2 x a x b)

= c2 + 2ab

Since the same answer must be obtained whichever method is used:

a2 + b2 + 2ab = c2 + 2ab

⇒ a2 + b2 = c2

Therefore, as a and b are two sides of a right angle triangle and c is the hypotenuse, this shows Pythagoras' Theorem.

Answer 3

The definition: a formula to find the hypnotuse of a right angle triangle named and invented by a greek mathemititon