 Hello friends, welcome to the session. I am Malka. Let's discuss the given question. A, B, C, D is a rectangle formed by the points A with the coordinates minus 1, minus 1, B, coordinates minus 1, 4, C, 5, 4 and D with the coordinates 5 and minus 1. P, Q, R, S are the midpoints of A, B, C, C, D and D, A respectively. Is the quadrilateral P, Q, R, S a square, a rectangle or a rhombus? Justify your answer. Here is the figure according to the question where A, B, C, D is a rectangle and P, Q, R, S are the midpoints of the sides A, B, C, C, D and D, A respectively. So, we have to find out whether the given quadrilateral P, Q, R, S is a square, a rectangle or a rhombus. Let's begin with the solution. Now, since we have to find out whether P, Q, R, S is a square, a rectangle or a rhombus. So, for this, we have to find out their sides and to calculate their sides, we have to find out the coordinates of P, Q, R and S. So, we find the coordinate of P, Q, R and S by the midpoint formula that is x1 plus x2 upon 2 and pi1 plus pi2 upon 2. So, coordinates of P are minus 1 plus minus 1 upon 2 and minus 1 plus 4 upon 2. So, coordinates of P are minus 1 and 3 by 2. Now, the coordinates of P are again we will use the same formula x1 plus x2 by 2 and pi1 plus pi2 by 2. So, coordinates of P are minus 1 plus pi by 2 and 4 plus 4 by 2. So, that is 4 by 2 and 8 by 2 or we can say 2 and 4 are the coordinates of the point Q. Now, we will find the coordinates of the point R, coordinates of R minus 1 plus 4 by 2 which is equal to 5 and 3 by 2. So, coordinates of R are 5 and 3 by 2. Now, coordinates minus 1 plus 5 by 2 and minus 1 plus minus 1 by 2 minus 1. So, coordinates of S are 2 and minus 1. Now, we are having the coordinates of the point P, Q, R and S. Therefore, now we will find all the 4 sides of the coordinate in P, Q, R, S. So, using distance formula, we will find all the 4 sides of P, Q, R, S. Therefore, P, Q equal to, now we will find P, Q. We are having the coordinates of P as minus 1 and 3 by 2, Q with 2 and 4. So, P, Q equal to, now here we are going to use the distance formula. So, P, Q equal to minus 1 minus 2 squared plus 3 by 2 minus 4 squared. This is equal to square root of 9 plus 25 by 4. Now, on taking LCM as 4, we get 56 plus 25 by 4. This is equal to square root of 61 by 2. Therefore, P, Q equal to square root of 61 by 2. Now, we will find Q, R. Therefore, Q, R equal to, again we are going to use the distance formula. Square root of 2 minus 5 squared plus 4 minus 3 by 2 squared. This is equal to square root of 9 plus 25 by 4. This is equal to square root of 61 by 2. Now, therefore we can say that Q, R equal to square root of 61 by 2. Now, we will find RS, therefore RS equal to square root of 5 minus 2 squared plus 3 by 2 plus 1 squared. This is equal to square root of 9 plus 25 by 4. Or this is equal to square root of 61 by 2. Therefore, RS is also square root of 61 by 2. Now, we will find SP. Therefore, SP equal to square root of 2 plus 1 squared plus minus 1 minus 3 by 2 squared. This is equal to square root of 9 plus 25 by 4. This is equal to square root of 61 by 2. Now, we are having all the sides of the quadrilateral PQ, RS and we see that all the sides are square root of 61 by 2. Now, we find the diagonals of the quadrilateral PQ, RS that is SQ and RP. So, diagonals, so first of all let us find RP equal to, we will use the distance formula to find RP, which is given by minus 1 minus 5 squared plus 3 by 2 minus 3 by 2 squared. So, this is equal to 6. Therefore, RP is equal to 6. Now, we find SQ. SQ equal to square root of 2 minus 2 squared plus 4 plus 1 squared. This is equal to 5. Here we see that all the sides of the quadrilateral are equal but diagonals are not equal. That is, RP is not equal to SQ. So, as we all know that in the rhombus all the sides are equal but diagonals are not equal. Therefore, we can say that the quadrilateral PQ, RS is rhombus. So, we can say that all 4 sides equal but diagonals not equal. Therefore, it is a rhombus. Therefore, PQ, RS is a rhombus. So, hope you understood the solution and enjoyed the session. Goodbye and take care.