 Hello and welcome to the session. In this session we are going to discuss similar triangles and slopes. In this session we will discuss with the help of similar triangles that slope is same between any two distinct points on a non-vertical line in the coordinate plane. Let us recall that two triangles are similar when they have same shape but size varies. Or we can say that corresponding sides are proportional and corresponding angles are same. Also if we have two distinct points X1, Y1 and X2, Y2 on a straight line in coordinate plane then slope of line is given by change in Y upon change in X which is equal to change in Y is given by Y2 minus Y1 upon change in X that is X2 minus X1. So slope is equal to Y2 minus Y1 upon X2 minus X1 and we should note that slope can be positive or negative. Now let us see the following graph. We have a non-vertical straight line. Now let us consider a few points on this line. That is minus 1 minus 5, 1 minus 1 and 4 5. Now if we draw a horizontal line from 1 minus 1 a vertical line from 4 5 we see they form a right angle triangle at the point where they meet. Similarly when we draw a horizontal line from minus 1 minus 5 a vertical line from 1 minus 1 they also form a right angle triangle at the point where they meet. Now we name the two triangles as A, B, C and C, P, R and we can see that both triangles have same shape that different size so the two triangles are similar to each other. So we can say that triangle A, B, C and triangle C, P, R are similar triangles. Now let us find the slope. Let us consider the points minus 1 minus 5 and 1 minus 1 and we know that slope is given by change in y upon change in x and change in y will be given by minus 1 minus of minus 5 upon change in x which is equal to 1 minus of minus 1 and this is equal to minus 1 plus 5 that is 4 upon 1 plus 1 that is 2 so we have slope as 2. If we take the points 1 minus 1 and 4 5 will be equal to change in y that is 5 minus of minus 1 upon change in x that is 4 minus 1 so we will get 5 plus 1 that is 6 upon 4 minus 1 that is 3 so slope is equal to 2 and hence we have seen that slope of a line is same then we take any two pairs of distinct points on the same straight line. Now we will discuss relationship between similar triangles and slope. Now here for the larger triangle that is C, P, R let us find the ratio of its vertical side to the horizontal side will be given by the vertical side R, P is equal to 1, 2, 3, 4, 5, 6 that is 6 units and the horizontal side C, P is given by 1, 2 and 3 that is 3 units so we have vertical side as 6 units and the horizontal side as 3 units so we get the ratio as 2 similarly for smaller triangle that is triangle ABC the ratio of vertical side to the horizontal side will be given by now the vertical side C, B is equal to 1, 2, 3 and 4 that is 4 units and the horizontal side AB is equal to 1 and 2 that is 2 units therefore we get vertical side that is 4 units by horizontal side that is 2 units so we get the ratio as 2 and we have also calculated the slope of the line as 2 and thus we notice that the ratio of sides of each similar triangle is equivalent to the slope and if the slope is negative then the ratio will be equivalent to the absolute value of the slope for example if slope is equal to minus 2 then its absolute value will be given by modulus of minus 2 which is equal to 2 then the ratio of sides of each similar triangle is equal to the absolute value because ratio of length of sides is never negative thus we can say that the relationship between similar triangles and slope is given by the simplified ratio of the vertical side length to the horizontal side length of each similar triangle formed by the slope of the line is equivalent to the absolute value of the slope so in this session we have learnt about the relationship between similar triangles and slope of a line this completes our session hope you enjoyed this session