 Hello and welcome to the session. I am Asha and I am going to help you with the following position which says show that if the diagonals of a quadrilateral bisect each other at right angles then it is a rhombus. Let us now begin with the solution and let A, B, C, D be the given quadrilateral. So we are given a quadrilateral A, B, C, D in which the diagonals bisect each other at right angles. That is if O is the point of intersection of these two diagonals then A O is equal to O C and D O is equal to O B. Also all these four angles are of 90 degree. That is A, C and B, D intersect at right angles. We have to show that quadrilateral A, B, C, D is a rhombus. Triangle A O D triangle A O B is equal to O A. This is common to both the triangles A O B is equal to angle A O D is equal to 90 degree. This is given to us that the diagonals bisect at right angles and also D O is equal to O B. This is also given to us. So in these two triangles we get that this side is common to both the triangles. This is given to us and these two angles are each of 90 degree. This is also given to us. Therefore by right hand side congruent criteria we find that triangle A O D is concurrent to triangle A O B. This further implies that A D is equal to A B since corresponding parts of congruent triangles are equal. So this is by CPCT. Similarly in triangle A O B and B O C we can show that A B is equal to B C and in triangle B O C and triangle C O D we can show that B C is equal to C D. So let this be equation number 1, this be 2 and this be 3. So from 1, 2 and 3 we get A B is equal to B C is equal to C D is equal to A D. So this implies all the sides of quadrilateral A B C D equal, quadrilateral A B C D is a rhombus. So this completes the session. Take care and bye for now.