 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says, A B CD is a rhombus and P Q R S R, the midpoints of site A B, P C, C D and D A respectively show that the correlator P Q R S is a rectangle. So interpreting the given question in the form of a figure it will look like this where A B CD is a rhombus and in a rhombus opposite side is a parallel also and equal also and we are given 4 points P Q R S such that P is the midpoint of A B, Q is the midpoint of B C, R is the midpoint of CD and S is the midpoint of D A and we have to show that P Q R S is a rectangle. So let's start with the solution and firstly let's write down what we are given. So we are given a rhombus A B CD in which P Q are midpoints of A B, C D and D A and we have to show that the correlator R S is a rectangle. Now to show that P Q R S is a rectangle, first we will show that P Q R S is a parallelogram. Now in triangle A D C points S and R midpoints of A D and D C respectively. So by midpoint theorem this implies that S R is parallel to AC and S R is equal to half of AC and this is by midpoint theorem which says that the right segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of it. So by midpoint theorem points P and Q are midpoints of sides A B and B C. So again by midpoint theorem this implies that P Q is parallel to AC and also P Q is equal to half of AC. So let this be one and this be two and from one and two we find that AC is a line which is parallel to both S R and P Q so this implies S R is parallel to P Q since two lines parallel to the same given line are parallel to each other and also S R is equal to half of AC and in turn half of AC is equal to P Q so this implies S R is equal to P Q and in a quadrilateral if a pair of opposite side is equal and parallel then it is a parallelogram and in quadrilateral P Q R S we have side S R equal to P Q and also S R is parallel to P Q so this implies that quadrilateral P Q R S is a parallelogram and now we will show that P Q R S is a rectangle. Now to show that it is a rectangle we will show that angle P is 90 degree since each angle of a rectangle is of 90 degree. So let us show that angle P is equal to 90 degree. Now let the point of intersection of the diagonals AC and B D B O this point be E and this point be F. Now since it is parallel to AC this implies P F is parallel to E O and also our midpoints of AB and AD so this implies SP is parallel to B D and SP is equal to half of B D. Now since SP is parallel to B D so this implies EP is parallel to let this be equation number three and this be equation number four in quadrilateral is equal to OF. PF is parallel to E O so this implies P is a parallelogram. Now we are given that AB CT is a rhombus this implies diagonals to each other at right angles so this implies this angle is equal which is angle E OF is 90 degree and does not we have proved that PF OE is a parallelogram so this implies angle EPF is equal to angle E OF since parallelogram equal so this implies angle is equal to 90 degree or we can say that angle P is equal to 90 degree and above we have shown that quadrilateral PQRS is a parallelogram and just now we have shown that angle P is 90 degree and therefore we have quadrilateral RS is a parallelogram in which angle P is equal to 90 degree therefore we can show that each angle of this quadrilateral is of 90 degree therefore this implies that quadrilateral RS is a rectangle so this completes the session take care and have a good day.