Question

prove that converse pythagoras theorem ​

Answer 1

Answer:

As in the diagram, ABC is a right-angled triangle

with right angle at C, then

a2 + b2 = c2

The converse of Pythagoras Theorem is:

If

a2 + b2 = c2 holds

then

DABC is a right angled triangle with

right angle at C.

Now construct another triangle as follows :

EF = BC = a

ÐF is a right angle.

FD = CA = b

In DDEF,

By Pythagoras Theorem,

……..(2)

By (1), the given,

Theorefore, AB = DE

But by construction, BC = EF

and CA = FD

D ABC @ D DEF (S.S.S.)

Answer 2

Converse of Pythagoras theorem is defined as: “If square of a side is equal to the sum of square of the other two sides then triangle must be right angle triangle”. Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is said that if this relation satisfies, then triangle must be right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy given condition a2 + b2 = c2, then the triangle is a right-angle triangle.

Statement: If the length of a triangle is a, b and c and c2 = a2 + b2, then the triangle is a right-angle triangle.

Converse of Pythagoras theorem

Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.

Converse of Pythagorean Theorem Proof

In △EGF, by Pythagoras Theorem:

EF2 = EG2 + FG2 = b2 + a2 …………(1)

In △ABC, by Pythagoras Theorem:

AB2 = AC2 + BC2 = b2 + a2 …………(2)

From equation (1) and (2), we have;

EF2 = AB2

EF = AB

⇒ △ ACB ≅ △EGF (By SSS postulate)

⇒ ∠G is right angle

Thus, △EGF is a right triangle.

Hence, we can say that the converse of Pythagorean theorem also holds.

Hence Proved.