Question

Write the converse of the Pythagoras Theorem and prove it.

Answer 1

Hint: In order to solve this problem we will apply the Pythagoras theorem which states that, in a right angle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. On drawing the figure and applying the pythagorean theorem you will prove this.

Complete step-by-step answer:
The converse of Pythagoras theorem can be stated as :
In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is at right angle.

Here we have drawn the diagram of the triangle ABC (figure (a)). The figure (b) is the figure of the same triangle knowing that angle O is of 90 degree, that has been drawn to prove that angle A is of 90 degree in triangle ABC.
We have given that ,
$B{C^2} = A{B^2} + A{C^2}$
We have to prove that angle A is 90 degrees.
In reference with figure (b),
Let OX be any ray
We can construct OY such that OY⊥OX
Let M ∈ OY such that OM = AC
Let N ∈ OX such that ON = AB
Draw MN, then WE CAN SAY
Triangle OMN is right angle triangle as OM⊥ON

And we also know ∠MON is a right angle
MN is the hypotenuse
According to the Pythagoras theorem

$M{N^2} = \,O{M^2} + O{N^2} = A{B^2} + B{C^2}$
But, $A{B^2} + B{C^2} = A{C^2}$
So, we can say that,
$M{N^2} = B{C^2}$
It means,
$MN = BC$
In Triangle ABC and △ONM consider the correspondence ABC↔ONM
AB≅ON (ON=AB)
AC≅OM (OM=AC)
BC≅MN (MN=BC)

The correspondence ABC ↔ ONM is congruence.
So Triangle ABC ↔ Triangle ONM (by SSS)
∠A≅∠O

But ∠O in Triangle ONM is right angle by construction
∠A is a right angle.
Hence, proved.

Note: To prove the converse of Pythagoras theorem we have drew a triangle of same length of sides but we have drawn the angle in second triangle 90 degree, then in the reference of it we have proved the converse of Pythagoras theorem and showed the angle as 90 degree.