 Hello and welcome to the session. I am Harsha and I am going to help you with the following question which says, A, B, C, D is a rectangle and P, Q, R, S are midpoints of sites A, B, C, D and D respectively show that polylateral P, Q, R, S is a rhombus. So first let us integrate the given question in the form of a figure which will look like this where A, B, C, D is the given rectangle and P, Q, R, S are the midpoints of sites A, B, C, C, D and D respectively and we have to show that P, Q, R, S is a rhombus. So let us start with the solution and we are given rectangle A, B, C, D in which P, Q, R and S are midpoints of sites A, B, B, C, D and D respectively and we have to show that polylateral P, Q, R, S is a rhombus and triangle D, A, B, S and P are midpoints of sites. So this implies S, P is parallel to D, B, P is equal to half of D, B. So let us pick question number one and now in triangle D, C, B points R and Q, B, C. So this implies Q is parallel to D, B and also R, Q is equal to half of D, B and let this be equation number two and therefore from one and two we have equal to R, Q since two lines parallel to the same given line are parallel to each other. So S, P is parallel to D, B and also R, Q is parallel to D, B. So this implies S, P is parallel to R, Q and also S, P is equal to half of D, B and in turn half of D, B is equal to R, Q. So this implies S, P is equal to R, Q. In a polylateral if a pair of opposite side is equal also and parallel also then it is a parallelogram. So this implies that R, S is a parallelogram. Since in this polylateral we have S, P parallel to R, Q and also S, P is equal to R, Q. So this implies P, Q, R, S is a parallelogram and we have to show that P, Q, R, S is a rhombus and to show that we will show that all the four sides of this polylateral are equal. So now we will show that P is equal to S, R. The single D, R is where side A, P is equal to D, R since A, P can be written as half of A, B. Since P is a midpoint of A, B we are given and A, B, C, D is a rectangle so A, B is equal to C, D. So replacing A, B by C, D and half of C, D is D, R. Since R is a midpoint of C, D. So let this be equation number 3 and also since S is a midpoint of A, D we are given. This implies that A, S is equal to S, D since S is a midpoint of A, D. A rectangle is of 90 degrees. Therefore this implies angle A is equal to angle D is equal to 90 degrees. So let this be equation number 4 and this will be 5. So from 3, 4 and 5 we have triangle A, P, S, concurrent to triangle, D, R, S and this is the side angle side axiom. Since these two triangles are concurrent this further implies that side P, S is equal to S, R. So let this be equation number 6. Similarly we can show that P, Q is equal to R, 2 and let this be equation number 7. R, S is a parallelogram so this implies the pair of opposite sides are equal and parallel therefore we have P, S equal to Q, R, S, R equal to P, Q and this is 8. Now from 6, 7 and 8 we get P, Q equal to Q, R is equal to R, S is equal to SP. So this implies all the sides are so parallelogram are equal. Therefore quadrilateral R, S is a rhombus. This is what we have to show. So this completes the solution. Take care and have a good day.