Question

how to prove converse of pythagoras theorum?

Answer 1

Converse of the Pythagorean Theorem:

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.



Proof:
Suppose the triangle is not a right triangle. Label the vertices A, B and C as pictured. (There are two possibilities for the measure of angle C: less than 90 degrees (left picture) or greater than 90 degrees (right picture).)



Construct a perpendicular line segment CD as pictured below.



By the Pythagorean Theorem, BD² = a² + b² = c², and so BD = c. Thus we have isosceles triangles ACD and ABD. It follows that we have congruent angles CDA = CAD and BDA = DAB. But this contradicts the apparent inequalities BDA < CDA = CAD < DAB AB < CAD = CDA < BDA

Answer 2

Converse of pythagorous theorem :in a triangle, if the square of one side is equal to the sum of the square of the other two sides, then the angle opposite to the side is a right angle.
Given :ABC is a triangle, ac^2 =ab^2 +bc^2
To prove: angleB=angleE=90°
Construction : construct a triangle DEF such that DE=AB, EF=BC and angleE=90°
Proof : by pythagorous theorem,
Df^2 =de^2 +ef^2
Df^2 =ab^2 +bc^2 [DE=AB, EF=BC]
Df^2 =ac^2
Df=ac------(1)
In triangle ABC and DEF,
AB=DE, BC=EF(by construction)
AC=DF(from equation.1)
By SSS cong. Criterion,
Triangle ABC is congruent to triangle DEF
=> angleB=angleE=90°(by cpct)
Hence, triangle ABC is a right angle triangle and right angled at B.


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