 Hello and welcome to the session. Let us discuss the following question. It says, without using the Pythagoras theorem, show that the points 4, 4, 3, 5 and minus 1, minus 1 are the vertices of a right-angled triangle. Let us understand the key idea behind this question. If two lines are perpendicular, then the product of their slopes is minus 1. So if m1 is the slope of one line and m2 is the slope of the other line, then m1 into m2 is equal to minus 1. So this knowledge will work as k idea. Let us now proceed on with the solution. We have to prove that these three are the vertices of right-tri-angle. So the given points are a 4, 4, b 3, 5 and c minus 1, minus 1. Now to prove this, these three are the vertices of right-tri-angle. We need to prove that ac is perpendicular to ab. That means we need to prove that product of the slope of ac and ab is minus 1. So we first find the slope of ab which we find by the formula y2 minus y1 upon x2 minus x1. Here y2 is 5, y1 is 4 upon x2 minus x1 that is 3 minus 4 and this is equal to 1 upon minus 1 that is minus 1. Now we find the slope of the line ac and now this is given by minus 1 minus 4 that is y2 minus y1 upon x2 minus x1 that is minus 1 minus 4 and this is equal to minus 5 upon minus 5 that is equal to 1. Let us call slope of ab as m1 and slope of ac as m2. Now m1 into m2 that is product of the slopes is equal to minus 1 into 1 that is equal to minus 1. Now since the product of the slopes is minus 1 therefore ac is perpendicular to ab. Hence triangle abc is a right-angled triangle. So that is all for this session. Goodbye and take care.