 Hello and welcome to the session. The question says show that the points A, B, C and D are vertices of a square. So to show that these four points are vertices of a square, we shall be showing that the distance between AB, BC, CD and DA is equal and also the length of diagonal is BD and AC is equal and this we shall be showing with the help of distance formula which says if P and Q are two points with coordinates x1, y1 and Q with coordinates x2, y2 then the distance PQ is given by root over x2 minus x1 whole square plus y2 minus y1 whole square. So with the help of this formula we shall be finding the distance between these points. So let's begin with the solution. Here we are given points A with coordinates 5, 6, B with coordinates 1, 5, C with coordinates 2, 1 and D with coordinates 6, 2 and we have to show that these are vertices of a square. We have to show that A, B, C and D are vertices of a square that is we have to show that the length of line segment AB is equal to BC is equal to CD is equal to DA and we have to show that AC is equal to BD. Let us find the distance between A and B. So this is equal to root over 1 minus 5 whole square plus 5 minus 6 whole square and this is equal to root over 16 plus 1 and this is equal to root over 17. The distance between BC is given by root over 2 minus 1 whole square plus 1 minus 5 whole square thus gives 1 square is 1 and minus 4 square is 16. So this is again equal to root over 17. Now let us find CD so this is equal to root over 6 minus 2 whole square plus 2 minus 1 whole square and this is equal to root over 16 plus 1 which is equal to root over 17 and lastly we shall find the distance DA and this is equal to root over 6 minus 5 whole square plus 2 minus 6 whole square and this is equal to root over 1 plus 16 which is equal to root over 17. So this shows that AB is equal to BC is equal to CD is equal to DA is equal to root over 17 units. Now let us find the length of the diagonals AC and BD. Now AC is equal to root over and this 5 whole square plus 1 minus 6 whole square and this is equal to root over 9 plus 25 which is equal to root over 34 and the distance BD is given by root over 6 minus 1 whole square plus 2 minus 5 whole square and this is again equal to root over 25 plus 9 and this is equal to root over 34. So this implies the length of the diagonals that is AC is equal to BD is equal to root over 34 sorry this is 34 and therefore we can say that A, B, C and D are vertices of a square since a quadrilateral is a square if all its sides are equal and its diagonals are equal so this completes the session by and take you.