 Hello and welcome to the session. Let's discuss the following question. It says show that the points a12b54c3a to b-16 are the vertices of a square. So let's now proceed on with the solution. And before moving on to the solution, we need to know the property of the square. It says that all the four sides of a square are equal. So the points a12b54c3a and d-16 are the vertices of a square. And only if a-b is equal to b-3, that is all the four sides are equal, is equal to c-d is equal to d-a. That is the distance between the points of a and b is equal to the distance between the points of b and c. And similarly is equal to the distance between the points of c and d and d and a. So now we need to write the distance formula, the distance formula for the points x1, y1 and qx2, y2 is given by under the root x2-x1 whole square plus y2-y1 whole square. So this is the distance between the points p and q. So p-q is given by this formula. So using this formula will prove that all the four sides are equal. So now we can find the distance between a and b. x1 is 1, y1 is 2, x2 is 5, y2 is 4. So it becomes 5-1 whole square that is x2-x1 whole square plus y2-y1 whole square and here y2 is 4, y1 is 2. So this is equal to under the root 4 square plus 2 square. This is again equal to 16 plus 4. So this is equal to root 20 and root 20 can be written as 2 into 2 into 5. So this can be written as 2 root 5. Now we will prove that bc is also 2 root 5. Here x1 is 5, y1 is 4, x2 is 3 and y2 is 8. So it becomes 3-i whole square plus 8-4 whole square. 3-5 is minus 2 whole square is 4 and 8-4 is 4, 4 whole square is 16. And again this is root 20 which is equal to 2 root 5. Now we find the distance between the points c and d. Now here the coordinates of c are 3 and 8 minus 1, 6. So cd becomes x2-x1 whole square that is minus 1, minus 3 whole square plus y2-y1 whole square that is 6-8 whole square. Minus 1 minus 3 is minus 4, minus 4 square is 16 and 6 minus 8 is minus 2, minus 2 square is 4. So again this is root 20 which is equal to 2 root 5. Now we have to find the distance between the points d and a that is we have to find the length of the side dA. And d is minus 1, 6, a is 1, 2. So here it becomes x2-x1 that is 1 minus minus 1 whole square plus y2-y1 that is 2 minus 6 whole square. This becomes 1 plus 1 whole square plus minus 4 whole square. So this is again 2 square plus 4 square is 16. So this is equal to 2 root 5. So we have proved that AB is equal to BC is equal to CD is equal to dA is equal to 2 root 5. So this implies given points are the vertices per square. So this completes the question and the session. Bye for now. Take care. Have a good day.