Question

converse of Pythagoras theorem state and prove

Answer 1

       What is Pythagoras Theorem ?

 

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.



How to prove The converse of Pythagoras Theorem ?  

Now construct another triangle as follows :

 

      EF = BC = a

ÐF is a right angle.

   FD = CA = b

 
   In  DDEF,

   By Pythagoras Theorem,

   By (1), the given,

   Theorefore,                 AB = DE

   But by construction,       BC = EF

   and                                 CA = FD

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


Lesson

 

We use Pythagoras Theorem and the concept of congruent triangles to prove the Converse of Pythagoras Theorem.

 

Answer 2

Explanation:

Statement:

In a Triangle the square of longer side is equal to the sum of squares of the other two sides, then the triangle is a right angled triangle.

Given -

A Triangle ABC such that

BC² = AB² + AC²

To Prove -

Angle A = 90°

Construction -

Draw a ∆DEF such that AB = DE and AC = DF and Angle D = 90°

Proof -

In ∆ABC,

BC² = AB² + AC² - Given

In ∆ DEF

EF² = DE² + DF²

Therefore,

EF² = AB² + AC²

(Since AB = DE, AC = DF)

Therefore,

BC² = EF² ie - BC = EF

Now, In ∆ABC and ∆DEF

AB = DE - By Construction

AC = DF - By Construction

BC = EF

Therefore

∆ABC ≅ ∆DEF by SSS test.

Thus,

Angle A = Angle D - CPCT

But, Angle D = 90° ( As per construction)

Therefore

Angle A = 90°

Hence Proved! .........