 Hi friends, I am Purva and today we will discuss the following question. Show that the points 7, 10, minus 2, 5 and 3 minus 4 are the vertices of an isosceles right triangle. Let us begin with the solution now. Now, let point A with coordinates 7, 10, point B with coordinates minus 2, 5 and point C with coordinates 3 minus 4 be the given vertices of triangle ABC. Now for any two points P and Q with coordinates x1, y1 and x2, y2 we have PQ is equal to under root of x2 minus x1 whole square plus y2 minus y1 whole square. Therefore here we have AB is equal to under root of now here x2 is equal to minus 2 and x1 is equal to 7 so we get minus 2 minus 7 whole square plus here y2 is equal to 5 and y1 is equal to 10 so we get 5 minus 10 whole square. So we get distance AB is equal to under root of minus 2 minus 7 whole square plus 5 minus 10 whole square units. Now this is equal to under root of minus 2 minus 7 is minus 9 whole square plus 5 minus 10 is minus 5 whole square units and we get this is equal to under root of 81 plus 25 units. This is further equal to under root of 106 units. We mark this as 1. Now AC is equal to under root of 3 minus 7 whole square plus minus 4 minus 10 whole square units. This is equal to under root of now 3 minus 7 is equal to minus 4 whole square plus minus 4 minus 10 is equal to minus 14 whole square units. This is equal to under root of 16 plus 196 units which is further equal to under root of 212 units. We mark this as 2 and we have BC is equal to under root of 3 minus minus 2 that is 3 plus 2 whole square plus minus 4 minus 5 whole square. This is equal to under root of 3 plus 2 is equal to 5 so 5 whole square plus minus 4 minus 5 is equal to minus 9 so minus 9 whole square units. This is further equal to under root of 25 plus 81 units which is equal to under root of 106 units. We mark this as 3. Now from 1 and 3 we see that AB is equal to BC therefore we have triangle ABC is isosceles triangle because we know that in an isosceles triangle two sides are equal and here we have AB is equal to BC therefore triangle ABC is isosceles triangle. Also we have AB square plus BC square is equal to now AB is equal to under root of 106 so we have AB square is equal to 106 and BC is equal to under root of 106 so BC square is equal to 106. So here we get AB square plus BC square is equal to 106 plus 106 and this is equal to 212 which is equal to AC square because we have AC is equal to under root of 212 therefore by converse of Pythagoras theorem which states that if in a triangle square of one side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is a right angle. We have triangle ABC is right angle that B. Hence we have the given points are vertices of an isosceles right angled triangle. This is our answer. Hope you have understood the solution. Bye and take care.