Question

State the converse of Pythagorean Theorem.

Answer 1

Hint: Write Pythagoras Theorem and then use the fact that to write converse of any statement, interchange its hypothesis and conclusion to write the converse of Pythagoras Theorem. Also, prove that the converse of Pythagoras Theorem holds using the congruency of triangles.

Complete step-by-step answer:
We have to state the converse of Pythagoras Theorem.
We will firstly state the Pythagoras Theorem and then state its converse.
Pythagoras Theorem gives the relation between lengths of sides of a right angled triangle. It stated that in a right angled triangle, with legs of length a and b and hypotenuse of length c, the square of length of hypotenuse is equal to the sum of square of length of each of the legs, i.e., \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\].
We will now state the converse of Pythagoras Theorem.
Converse of Pythagoras Theorem says that if the square of the length of the longest side of a triangle is equal to the sum of the square of the other two sides, then the triangle is a right angled triangle, i.e., in a triangle \[\vartriangle ABC\], if \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] holds where c is the length of longest side and a and b are length of other two sides, then \[\angle C\] is right angled, as shown in the figure.

We will now prove the converse of Pythagoras Theorem.
We will prove it by contradiction. Let’s assume that in \[\vartriangle ABC\], \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] holds but the triangle is not right angled.
We will construct another triangle \[\vartriangle PQR\] such that \[PQ=a,QR=b\] and \[\angle Q\] is right angled, as shown in the figure.

By Pythagoras Theorem, we have \[{{\left( PR \right)}^{2}}={{a}^{2}}+{{b}^{2}}\].
We know that \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\].
Thus, we have \[{{\left( PR \right)}^{2}}={{a}^{2}}+{{b}^{2}}={{c}^{2}}={{\left( AB \right)}^{2}}\]. So, we have \[PR=AB\].
We observe that \[\vartriangle PQR\] is congruent to \[\vartriangle ABC\] as all the three sides of both the triangles are equal.
Since, both the triangles are equal and \[\vartriangle PQR\] is a right angled triangle, \[\vartriangle ABC\] must be right angled as well.
Thus, our assumption is wrong.
Hence, \[\vartriangle ABC\] is a right angled triangle given \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] holds.

Note: One must clearly know Pythagoras Theorem to state its converse. We also need to know the fact that to write the converse of a statement, we interchange the hypothesis and conclusion of the statement. Otherwise, we won’t be able to solve this question. Also, we need to know the definition of congruency and the conditions required to prove that two triangles are congruent to each other to prove the converse of Pythagoras Theorem.