 Hello and welcome to the session. In this session we discussed the following question which says in figure that is this figure P and Q are points on the sides AB and AC respectively of a triangle ABC such that AP is equal to 3.5 centimeters, PB equal to 7 centimeters, AQ equal to 3 centimeters and QC equal to 6 centimeters. If PQ equal to 4.5 centimeters point BC. 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 ABC we have AP upon PB is equal to 3.5 upon 7 which is equal to 1 upon 2 then AQ upon QC is equal to 3 upon 6 which is equal to 1 upon 2 that is we have AP upon PB is equal to AQ upon QC so we say that the line PQ divides the two sides AB and AC of triangle ABC in the same ratio therefore this implies that the line PQ is parallel to line BC. Now in triangle ABC and triangle APQ we have angle A is equal to angle A since it is the common angle then angle APQ is equal to angle ABC since they are the corresponding angles and we have PQ is parallel to BC then angle AQP is equal to angle ACB since they are the corresponding angles and PQ is parallel to BC Thus we have the triangle ABC is similar to triangle APQ by AAA similarity and thus we have that AB upon AP is equal to BC upon PQ is equal to AC upon AQ Now this AB is equal to AP plus PB upon AP is equal to BC upon PQ is equal to AC now AC is equal to AQ plus QC upon AQ We got this from this figure that is you can see that AC is equal to AQ plus QC and AB is equal to AP plus PB Now on substituting the values for AP PB and AP we get 10.5 upon 3.5 Equal to BC which we have to find out upon PQ which is of length 4.5 Now since we need to find the length of BC so we can ignore this part and we consider these two So from here we get BC 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 BC is equal to 13.5 centimeters Hence our final answer is BC equal to 13.5 centimeters So this completes the session hope you have understood the solution for this question