 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says verify the following minus 1, 2, 1, 1 minus 2, 5 and 4 minus 7, 8 and 2 minus 3, 4 are the vertices of a parallelogram. So let us start with the solution to this question. Now here we use a very important property of parallelogram that opposite sides of a parallelogram they are equal but its diagonals are not equal. So using this property let us find out the solution to this question. Let the point A be minus 1, 2, 1, the point B be 1 minus 2, 5, the point C be 4 minus 7, 8 and the point D be 2 minus 3, 4. Let these be the 4 given points. Now applying the formula that says that if a point P x1, y1, z1 and a point Q x2, y2, z2 are 2 points then the distance between them is given by square root of x2 minus x1 the whole square plus y2 minus y1 the whole square because z2 minus z1 the whole square. So using this let us find out the distance between the point A and B. So AB is equal to square root of 1 minus minus 1 the whole square plus minus 2 minus 2 the whole square plus 5 minus 1 the whole square. This is equal to square root of now 1 minus minus 1 is 1 plus 1, 1 plus 1 is 2 and square of 2 is 4 plus minus 2 minus 2 is minus 4 and square of minus 4 is 16, 5 minus 1 is 4 and square of 4 is again 16. So this is equal to square root of 36 that is equal to 6. Similarly we find out the distance between the point B and C. So BC will be equal to square root minus 1 the whole square plus minus 7 minus of minus 2 is minus 7 plus 2 that is minus 5 and square of minus 5 we have here plus 3 square this is equal to square root is 9 minus 5 square is 25 plus 9 that is equal to square root of 43. Similarly we find out the distance at the whole square minus 3 minus minus 7 the whole square the whole square that is equal to square root of minus 2 is 7 is 4 and square of 4 is 16 4 is again 16 that is equal to square root of 36 that is equal to 6. Finally we find out the distance between the point A and D. So AD is equal to this minus 1 is 2 plus 1 the whole square plus minus 3 minus 2 the whole square plus 4 minus 1 the whole square that is equal to plus 1 is 3 square of 3 is 9 plus minus 3 minus 2 is minus 5 and square of minus 5 is 25 plus 4 minus 1 is 3 and square of 3 is 9 so we have square root of 9 plus 25 plus 9 that is equal to square root of 43. Let this be the parallelogram ABCD now we have AB is equal to 6 units BC is root 43 units and AD is root 43 units. Now we see that opposite sides of this parallelogram are equal because AB is equal to CD and BC is equal to AD so it is either a rectangle or it is a parallelogram now we find out AC that is this diagonal and we find out BD that is this diagonal. So AC is the distance between points A and C so AC will be equal to square root of 4 minus minus 1 the whole square plus 7 minus 2 the whole square plus 8 minus 1 the whole square this is equal to 4 plus 1 the whole square is 5 square that is 25 plus minus 7 minus 2 is minus 9 and square of minus 9 is 81 plus 8 minus 1 is 7 and square of 7 is 49 under the square root this is equal to square root of 155 and we also see that BD is equal to square root of 2 minus 1 the whole square plus minus 3 minus of minus 2 is minus 3 plus 2 the whole square plus 4 minus 5 the whole square that is equal to square root of 1 square is 1 plus minus 1 square is 1 plus minus 1 square is again 1 that is equal to square root of 3 so we see that diagonal is not equal to diagonal this diagonal is not equal to this diagonal therefore ABCD is a parallelogram so first of all by proving that opposite sides are equal we show that this is either a parallelogram or a rectangle but we know that in a rectangle the diagonals are of equal proof that diagonals are not equal to each other therefore we can say that is a parallelogram so this was what we were supposed to prove in this question I hope that you understood the question and enjoyed the session have a good day