 Hello and welcome to the session. In this session, we discussed the following question. It says, in figure, that is this figure, P and Q are points on the sides A, B and A, C respectively of a triangle A, B, C such that A, P is equal to 3.5 centimeters, P, B equal to 7 centimeters, A, Q equal to 3 centimeters and Q, C equal to 6 centimeters. If P, Q equal to 4.5 centimeters, point B, C. We know that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. This will be taken as the key idea for this question. Now, let's move on to the solution. In triangle A, B, C, we have A, P upon P, B is equal to 3.5 upon 7, which is equal to 1 upon 2. Then A, Q upon Q, C is equal to 3 upon 6, which is equal to 1 upon 2. That is, we have A, P upon P, B is equal to A, Q upon Q, C. So, we say that the line P, Q divides the two sides A, B and A, C of triangle A, B, C in the same ratio. Therefore, this implies that the line P, Q is parallel to line V, C. Now, in triangle A, B, C and triangle A, P, Q, we have angle A is equal to angle A since it is the common angle. Then angle A, P, Q is equal to angle A, B, C since they are the corresponding angles and we have P, Q is parallel to B, C. Then angle A, Q, P is equal to angle A, C, B since they are the corresponding angles and P, Q is parallel to B, C. Thus, we have the triangle A, B, C is similar to triangle A, P, Q by A, A, A similarity and thus we have that A, B upon A, P is equal to B, C upon P, Q is equal to A, C upon A, Q. Now, this A, B is equal to A, P plus P, B upon A, P equal to B, C upon P, Q equal to A, C. Now, A, C is equal to A, Q plus Q, C upon A, Q. We got this from this figure that is you can see that A, C is equal to A, Q plus Q, C and A, B is equal to A, P plus P, B. Now, on substituting the values for A, P, P, B and A, P we get 10.5 upon 3.5 equal to B, C which we have to find out upon P, Q which is of length 4.5. Now, since we need to find the length of B, C so we can ignore this part and we consider these two. So, from here we get B, C is equal to 10.5 into 4.5 upon 3.5. Now, from here we get that 5, 7 times is 35, 5, 9 times is 45, then we have 7, 15 times is 105. Then this is equal to 135 upon 10 which is equal to 13.5. So, we get that B, C is equal to 13.5 centimeters. Hence, our final answer is B, C equal to 13.5 centimeters. So, this completes the session. Hope you have understood the solution for this question.