 Hello and welcome to the session. In this session we will discuss the question which says that show that a64b5-2 and c7-2 are the vertices of an isosceles triangle also find the length of the median through a. Now before starting the solution of this question we should know our result and that is if the coordinates of the point a are given by x1, y1 and the coordinates of the point v are given by x2, y2 then by distance formula av is equal to x2 whole square plus y1 minus y2 whole square. Now this result will work out as a key idea for solving out this question and now we will start with the solution. Now given a triangle abc in which the coordinates of a point b minus 2 and the coordinates of c are 7 minus 2 proof that this triangle is an isosceles triangle. Now for this we will find the length of all the sides by using the distance formula that is this formula which is given in the key idea and if the length of any two sides will be equal then the triangle is isosceles. Now let us take the coordinates of a as x1, y1 the coordinates of b as x2, y2 and the coordinates of c as x3, y3. Now by distance formula this ab is equal to square root of 2 whole square plus y1 minus y2 whole square. Now putting the values of x1, y1 and x2, y2 here this implies ab is equal to square root of 5 whole square plus this of minus 2 which is equal to square root of which is further equal to square root of 1 plus 36 which is equal to root 37. Therefore the length of the side ab is equal to root 37 units. The length of the side bc will be equal to square root of x3 whole square plus y2 minus y3 whole square. Now putting the values of x2, y2 and x3, y3 here this implies bc is equal to square root of minus 7 whole square minus of minus 2 whole square equal to square root of minus 2 whole square plus 0 whole square which is equal to root 4 which is further equal to 2. Therefore the length of the side bc is equal to 2 units. Now again by distance formula will be equal to square root of whole square plus y1 minus y3 whole square. Now putting the values of x1, y1 and x3, y3 here this will be equal to square root of x minus 7 whole square plus 4 minus of minus 2 whole square which is equal to square root of minus 1 square plus 6 which is further equal to square root of 1 plus 36 which is equal to root 37. Therefore ac is equal to root 37 units also root 37 units equal to ac is equal to root 37 units. This implies triangle a, b, c is an i and 2 sides of the triangle. The length of median through a numbered ad side bc at the point that d is the midpoint. Now the coordinate point formula plus x3 by 2 by 2 plus y3 by 2 as d is the midpoint of b. So this will be equal to putting the values of x2 by 2 and x3 by 3 here this will be 5 plus 7 by 2 and minus 2 minus 2 by 2 12 by 2 and minus 4 by 2 which is further equal to 6 and minus 2 of d. So the coordinates of d are 6 minus 2 and let us take it as x4 by distance formula this ad which is the length of the median is equal to square root of x4 whole square minus y4 whole square which is equal to square root of which is equal to square root of 0 square which is equal to root 36 which is equal to 6. Therefore the median ad is equal to 6 units. Now the given question and that is all for this session hope you all have enjoyed this session.