Question

To get 100points!!!!!!... heres the question
PROOF OF PYTHAGORAS THEOREM BY DIFFERENT METHODS
(AT LEAST 5 METHODS)
PLZ ITS URGENT...

Answer 1

ANSWER:--------

Pythagorean theorem. If, in a triangle, angles α, β, γ lie opposite the sides of length a, b, c, then(EWD) sign(α + β - γ) = sign(a² + b² - c²),where sign(t) is the signum function:sign(t)= -1, for t < 0,sign(0)=  0,sign(t)=  1, for t > 0.The theorem this page is devoted to is treated as "If γ = p/2, then a² + b² = c²."

hope it helps:-----

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Answer 2

Bhaskara's First Proof

Bhaskara's proof is also a dissection proof. It is similar to the proof provided by Pythagoras. Bhaskara was born in India. He was one of the most important Hindu mathematicians of the second century AD. He used the following diagrams in proving the Pythagorean Theorem.

In the above diagrams, the blue triangles are all congruent and the yellow squares are congruent. First we need to find the area of the big square two different ways. First let's find the area using the area formula for a square.

Thus, A=c^2.

Now, lets find the area by finding the area of each of the components and then sum the areas.

Area of the blue triangles = 4(1/2)ab

Area of the yellow square = (b-a)^2

Area of the big square = 4(1/2)ab + (b-a)^2

= 2ab + b^2 - 2ab + a^2

= b^2 + a^2

Since, the square has the same area no matter how you find it

A = c^2 = a^2 + b^2,

concluding the proof.

Bhaskara's Second Proof of the Pythagorean Theorem

In this proof, Bhaskara began with a right triangle and then he drew an altitude on the hypotenuse. From here, he used the properties of similarity to prove the theorem.

Now prove that triangles ABC and CBE are similar.

It follows from the AA postulate that triangle ABC is similar to triangle CBE, since angle B is congruent to angle B and angle C is congruent to angle E. Thus, since internal ratios are equal s/a=a/c.

Multiplying both sides by ac we get

sc=a^2.

Now show that triangles ABC and ACE are similar.

As before, it follows from the AA postulate that these two triangles are similar. Angle A is congruent to angle A and angle C is congruent to angle E. Thus, r/b=b/c. Multiplying both sides by bc we get

rc=b^2.

Now when we add the two results we get

sc + rc = a^2 + b^2.

c(s+r) = a^2 + b^2

c^2 = a^2 + b^2,

concluding the proof of the Pythagorean Theorem.

First, we need to find the area of the trapezoid by using the area formula of the trapezoid.

A=(1/2)h(b1+b2) area of a trapezoid

In the above diagram, h=a+b, b1=a, and b2=b.

A=(1/2)(a+b)(a+b)

=(1/2)(a^2+2ab+b^2).

Now, let's find the area of the trapezoid by summing the area of the three right triangles.

The area of the yellow triangle is

A=1/2(ba).

The area of the red triangle is

A=1/2(c^2).

The area of the blue triangle is

A= 1/2(ab).

The sum of the area of the triangles is

1/2(ba) + 1/2(c^2) + 1/2(ab) = 1/2(ba + c^2 + ab) = 1/2(2ab + c^2).

Since, this area is equal to the area of the trapezoid we have the following relation:

(1/2)(a^2 + 2ab + b^2) = (1/2)(2ab + c^2).

Multiplying both sides by 2 and subtracting 2ab from both sides we get

a^2+b^2=c^2

concluding the proof.