 Hi and welcome to the session. Let us discuss the following question. Question says in figure 6.56 PS is the bisector of angle QPR of triangle PQR. Prove that QS upon SR is equal to PQ upon PR. This is the figure 6.56. Let us now start with the solution. We are given in triangle PQR PS is the bisector of angle QPR. That is angle QPS is equal to angle RPS. Now we have to prove that QS upon SR is equal to PQ upon PR. Now before starting the required proof let us do one construction. Through R draw a line parallel to PS intersecting QP produced at T. Let us now start the required proof. Now we know PS is parallel to RT and QT is transversal. So angle QPS is equal to angle PTR. Clearly we can see these two angles are corresponding angles. And we know corresponding angles are equal when two parallel lines are intersected by a transversal. So we can write PS is parallel to TR and QT is transversal. So angle QPS is equal to angle PTR. Now again we can see PS is parallel to TR and PR is transversal. So angle SPR is equal to angle PRT. We can see these two angles are alternate angles so they are equal. So we can write PS is parallel to TR and PR is transversal. So angle SPR is equal to angle PRT. We also know that PS bisects angle QPR that is angle QPS is equal to angle SPR. This is given in the question. Now in these two expressions we know these two angles are equal. So this implies angle PTR is equal to angle PRT. Let us name this expression as 1, this expression as 2 and this expression as 3. So we can write from expressions 1, 2 and 3 we get angle PTR is equal to angle PRT. Now we get these two angles are equal. Now if we consider this triangle PRT then PT is equal to PR. We know in a triangle equal angles have equal opposite sides. So we can write in triangle PRT angle PTR is equal to angle PRT. This we have already shown above. So this implies PT is equal to PR. Now let us name this expression as 4. Now let us consider triangle QTR. In triangle QTR PS is parallel to RT. So by basic proportionality theorem we get QAS upon SR is equal to QP upon PT. So we can write in triangle QTR PS is parallel to RT by construction. Now by basic proportionality theorem we get QP upon PT is equal to QAS upon SR. We know basic proportionality theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. Now here in expression 4 we can see PT is equal to PR. So we can substitute PR for PT here. So we can write this implies QP upon PR is equal to QAS upon SR or we can write it as QAS upon SR is equal to PQ upon PR. We know QP can be written as PQ. So our required answer is QAS upon SR is equal to PQ upon PR. So this completes the session. Hope you understood the session. Take care and keep smiling.