 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 0 7 minus 10 1 6 minus 6 and 4 9 minus 6 are the vertices of an isosceles triangle. So this is what we need to verify that these three points are the vertices of an isosceles triangle. Let us see the solution. Now before starting we should know that triangle is isosceles if its two sides are equal. So let the point A be 0 7 minus 10, the point B be 1 6 minus 6 and the point C be 4 9 minus 6. Let these three be the given three points. Now we know that if a point P with coordinates x 1, y 1, z 1 and the point Q x 2, y 2, z 2 are two points then the distance between them is given by P Q is equal to square root of x 2 minus x 1 the whole square plus y 2 minus y 1 the whole square plus z 2 minus z 1 the whole square. So applying this formula we see that distance between A and B is given by A B is equal to square root of 1 minus 0 the whole square plus 6 minus 7 the whole square plus minus 6 minus 10 the whole square. This is equal to now square of 1 is 1, 6 minus 7 is minus 1 and square of minus 1 is 1 plus minus 6 minus 10 is minus 16 but here we have minus 6 minus minus 10 that means minus 6 plus 10 minus 6 plus 10 is 4 and square of 4 is 16 so plus 16 this is equal to square root of 18 which can also be written as 3 root 2. Now we find out the distance between the point B and C so using the same formula we see that B C is equal to square root of 1 or 4 minus 1 the whole square plus 9 minus 6 the whole square plus minus 6 minus of minus 6 that is minus 6 plus 6 the whole square this is equal to now square of 4 minus 1 is square of 3 and square of 3 is 9 so 9 plus 9 minus 6 is 3 and square of 3 is 9 plus minus 6 plus 6 is 0 so we have square root of 18 and this is equal to 3 root 2. Now we find out the distance between C and A so C is equal to square root of 4 minus 0 the whole square plus 9 minus 7 the whole square plus minus 6 minus of minus 10 is minus 6 plus 10 the whole square this is equal to square root of now square of 4 is 16 9 minus 7 is 2 square of 2 is 4 and square of 4 is 4 square of 4 square of 4 is 16 this is equal to square root of 36 and square root of 36 is 6. Now we see that as sum of 2 distances is not equal to third distance but A B is equal to B C so we say that they form a triangle that is isosceles because we know that sum of any 2 distances is not equal to third distance so A B C they are not collinear since A B is equal to B C so they form an isosceles triangle 4 we have proved that 0.07 minus 10 0.16 minus 6 or 9 minus 6 they are the vertices high angle so this is our answer to the question I hope that you understood the question and enjoyed the session have a good day.