 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, 10, minus 1, 6, 6 and minus 4, 9, 6 are the vertices of a right angle triangle. So this is what we need to verify that these three points are the vertices of a right angle triangle. Now we know that if we have a point P x1, y1, z1 and a point Q x2, y2, z2 then the distance between them is given by P Q is equal to square root of x2 minus x1 the whole square plus y2 minus y1 the whole square plus z2 minus z1 the whole square. So let us start with the solution to this question. Let the point A be 0, 7, 10 the point B be minus 1, 6, 6 and the point C be minus 4, 9, 6 be the three given points. So applying the distance formula we see that A B is equal to square root of minus 1 minus 0 the whole square plus 6 minus 7 the whole square plus 6 minus 10 the whole square. This is equal to square root of square of minus 1 is 1 plus 6 minus 7 is minus 1 and square of minus 1 is 1 plus 6 minus 10 is minus 4 and square of minus 4 is 16. So this is equal to 16 plus 1 plus 1 is 18 so square root of 18 or we can say that A B square is equal to 18. Here we have taken square on both the sides so A B square equals to 18. Now B C is equal to square root of minus 4 minus minus 1 is plus 1 so minus 4 plus 1 the whole square plus 9 minus 6 the whole square plus 6 minus 6 the whole square. This is equal to square root of minus 4 plus 1 is minus 3. Square of minus 3 is 9 plus 9 minus 6 is 3 square of 3 is 9 plus 6 minus 6 is 0. So this is equal to square root of 18 or B C square is equal to 18. Similarly we can find out C A. C A is equal to square root of minus 4 minus 0 the whole square plus 9 minus 7 the whole square plus 6 minus 10 the whole square. So let us just write down what we have seen. C A is equal to square root of minus 4 minus 0 the whole square plus 9 minus 7 the whole square plus 6 minus 10 the whole square. This is equal to square root of square of minus 4 is 16. 9 minus 7 is 2 square of 2 is 4 plus 6 minus 10 is minus 4 and square of minus 4 is 16. This is equal to square root of 36. Now taking square on both the sides we have C A square is equal to 36. Therefore we note here that C A square is equal to AB square plus BC square now using the converse of Pythagoras theorem that states that if square of one of the sides is equal to sum of square of other two sides then the triangle is right-angled. Therefore we can say that ABC is a right-angled triangle. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.