 Hello and welcome to the session. I am Asha and I am going to help you with the following problem that says in figure 9.16 p is a point in the interior of a parallelogram A, B, C, D show that first area of A, B, B plus area of B, C, D is equal to half area of A, B, C, D. Second is area of A, B, D plus area of P, B, C is equal to area of A, B, B plus area of P, C, D. We have also given an hint that through P draw a line parallel to A, B. So let this be the line which is drawn parallel to A, B such that it intersects A, D at a point E and we see at a point F. So we are given a point P in the interior of parallelogram A, B, C, D and first we have to show that area of A, B, B plus area of P, C, D is equal to half area of A, B, C, D. Since we are given that A, B, C, D is a parallelogram this implies P is parallel to C, D and also by construction we have drawn E, F parallel to A, B. So this implies that E, F is also parallel to C, D since if two lines are parallel to the same given line then they are parallel to each other and they are the line C, D and E, F are parallel to the same line A, B so they both are parallel to each other. Now first let us learn a simple property which says if a triangle in the parallelogram same base between the same parallel lines of the area of parallelogram that area of P, B is equal to half area of A since the triangle A, P, B and A, E, F, B which is a parallelogram since E, F is parallel to A, B and A is parallel to B, F are on the same base A, B and between the same parallel lines A, B and E, F so the area of triangle will be half the area of parallelogram. Let this be equation number one similarly area of D, P, C is equal to half area of D, E, F, C. Next equation number two. Now on adding one and two on the left hand side we have area of A, P, B plus area of D, P, C and on the right hand side we have taking half area of A, E, F, B plus area of D, E, F, C. Now the sum of areas of these two parallelograms is equal to the area of A, B, C, D so we have half area of A, B, C, D thus we have proved the first part which is area of A, P, B plus area of D, P, C is equal to half area of A, B, C, D so this completes the first part. Now next we have to prove that area of A, P, D plus area of P, B, C is equal to area of A, P, B plus area of P, C, D. Now let us draw a line through P which is parallel to B, C which is our construction draw G, H parallel to B, C. Now since A, B, C, D is a parallelogram so this implies A, D is parallel to B, C since opposite sides of parallelogram are equal and also since B, C is parallel to G, H so this implies that A, D is parallel to G, H. Now by using the above property which says if a triangle and a parallelogram are on the same base and between the same parallel lines then area of triangle is equal to half area of parallelogram therefore we can say that area of triangle A, P, D is equal to half area of parallelogram A, G, H, D. Now A, G, H, D is a parallelogram since we are given that A, B is parallel to D, C since A, B, C, D is a parallelogram so this implies A, G is parallel to D, H and also by construction we have A, D parallel to G, H. Just now we have proved and in a correlator if opposite sides are parallel then it is a parallelogram so A, G, H, D is a parallelogram thus we can apply the property similarly applying the property on triangle B, P, C and parallelogram H, C, B, G we have area of B, P, C is equal to half area of H, C, B, G. So let this be equation number 3 and this be equation number 4. Now in having equation 3 and 4 on the left hand side we have area of A, P, D plus area of B, P, C this is equal to half taking common and right hand side we have area of A, G, H, D plus area of H, C, B, G this can further be written as half area of A, B, C, D since the sum of parallelograms A, G, H, D and G, B, C, H is equal to A, B, C, D and thus the area of these two parallelograms is equal to the area of A, B, C, D. So we have area of A, P, D plus area of P, B, C is equal to half area of A, B, C, D let this be equation number 5. Now in the first part we prove that area of A, P, B plus area of P, C, D is equal to half area of A, B, C, D and now since the right hand side of both these equations are equal so the left hand side are also equal thus we have area of A, P, D plus area of P, B, C is equal to area of A, P, B plus area of P, C, D so this is the second part we have to prove so this completes the second part take care and bye for now.