 Hi, and how are you all today? The question says, P and Q are respectively the midpoints of side A, B and P, C of a triangle A, B, C and R is the midpoint of A, B. Show that area of P, R, Q is half of area of A, R, C. Area of R, Q, C is equal to 3 by 8 area of A, B, C and area of P, B, Q is equal to area of A, R, C. So we will be proving all these three one by one and let's have a diagram that can model this situation. Now this is a diagram which satisfies all the condition which are given to us. P and Q are the midpoints of A, B and B, C respectively. R is the midpoint of A, P. And now let us start with the first proof. First part given to us is area of P, R, Q is equal to half of area of A, R, C. This is what we need to prove over here. We will be proving it very slowly and then only you will be able to understand this proof. Now let us start with the left hand side area of P, R, Q. Now you must know one of the theorems which says or properties which says that a median of a triangle divides a triangle into two equal areas. Now we can write this here, a median of a triangle divides it into two triangles first of all of equal area. Right? Now we will be making a lot more use of this property in solving this question. Now here we have started off with area of P, R, Q. This is P, R, Q. This is the triangle that we are talking about right now. That is, let us shade it a little and we need to prove that this is half of A, R, C and A, R, C is this triangle. Right? So let us start. Now we can say that P, R, Q, if you notice, R is the midpoint of A, P. So if you join R with Q, Q, R will become the median, isn't it? So this median will divide this triangle A, Q, P into two equal parts. Let me draw a small triangle that will help you to understand it more clearly. Now this is E, P, Q. We know that R is the midpoint of A, P. So if we join R with Q, Q, R will become the median of this triangle and we can say that area of P, R, Q is half of this whole big triangle. That is area of A, P, Q. Right? As we have discussed, a median divides a triangle into two equal half. So this is half of this whole big triangle. Further. Next we can say that we have this triangle. Now I will be drawing small triangles just roughly to make your understanding more clearer. Now we know that P is the midpoint of A, B. So if you join P with Q, it will become the median again and it will divide it into two equal half. So that means this area is equal to this area. So we can say that A, P, Q. A, P, Q that we have discussed in our first part is equal to P, B, Q. Right? So here we can write in place of A, P, Q, P, B, Q. Because this median is dividing it into two equal halves and the area of this triangle is equal to this triangle. So in place of this triangle we can write this triangle. Similarly, we know that if you notice it from above figure only, I am drawing this figure from above figures only. But since it is having a lot more triangles, I am taking those triangles which I am needing in every step. Now we have this triangle P, B, C and Q is the midpoint of B, C. If we join P, Q, if you notice P, Q, it is the median of triangle A, B, C. Now A, P, B, Q, this triangle can be written as it is half of P, B, C. Right? So we can write that it is half of area of P, B, C. Further, now P, B, C, P, B, C is equal to A, P, C. Let me draw a diagram to make you. Now this is a triangle, the initial triangle A, B, C which we were having and we know that P is the midpoint of A, B. So C, P will be the median. Now we can say that P, B, C is equal to A, P, C. The areas are equal. So in place of P, B, C in the above step, we can write down P, C. Right? Further, now A, P, C, this triangle, this is A, P, C. R is the midpoint and on joining C with R, this will become the median and this areas are equal. So in place of A, P, C, I can say that this is twice of A, R, C. Since it's a question, I want A, R, C. That is why I am moving from here to here in order to get A, R, C anywhere in the right hand side. So I can write it that A, P, C which was in the above step can be written as it is twice of area of A, R, C because it's the double of any of these parts. It can be twice of R, P, C also and A, R, C also. But we are taking A, R, C because in the right hand side of the proof, I want A, R, C. Now these two will get cancelled out and we are left with half of area of A, R, C which is equal to R, RHS. So here LHS is equal to RHS and hence we have proved the first part. Now let us start with the second part. Here we need to prove that area of RQC is equal to 3 by 8 area of A, B, C. Or we can write it that 8 times area of RQC is equal to 3 times area of A, B, C. This 8 is be taken on the left hand side. Now let us start with the left hand side. That is 8 times area of RQC. Now we can write area of RQC, RQC as 1 by 2 area of RBC. Let me draw a rough figure that will make it more clearer. Here we have RBC, a triangle. Q is the median. RQ will be the median. So this whole triangle that is RQC is half of this big triangle, isn't it? So I can write that this area of RQC is half of this whole big triangle which is RBC. On simplifying I have now here 4 times area of RBC. Further RBC can also be written as if you see RBC. Let me draw it and here I have B, this is RBC. P is not the midpoint. Now if you carefully see that area of RBC can be written as area of RPC, this plus area of this triangle that is RP or PB, right? This whole triangle is the sum of these two areas of the triangle. Now I can write RPC as RPC as half of this big triangle, half of area of APC and PVC, this is PBC. Can be written as this is equal to APC. So in place of PVC I can write APC that are equal. Now if you take APC common and add 1 by 2 in 1 as it is 1 over here, we have 3 by 2 area of APC. On simplifying we have 6 times area of APC. Now if you observe we have to have ABC. So APC can be written as this is half of the portion of this big triangle. So I can write that 6 times area of APC can be written as that this is half of ABC. On simplifying we have 3 times ABC which is our RHS and hence we have proved the second part. Now starting off with the third part we need to prove that area of P PQ is equal to area of ARC. Now here let us start with our right hand side. You can start with any of the sides. Area of ARC can be easily written as half of area of PCA. Let us have a diagram. APC R is the midpoint therefore CR will be the median. So ARC this triangle can be written as half of this big triangle that is APC or PCA you can name it as anything. So this is half of this triangle whose name is PCA. Now PCA can be written as ABC. Can you name it Y? Right exactly. I know that ABC is a triangle where P is here the midpoint. So PCA this triangle can be written as equal to PBC as they both are equal in area. So I can write any for I can write PCA or PBC they both are equal areas further. Now this PBC is twice the area of P B Q. How? I have this triangle PBC and Q is the midpoint. Now here if you see PBC that is this big triangle can be written as twice of this triangle as twice of this triangle but I want PBQ that is why I have written twice of PBQ on simplifying it becomes equal to area of PBQ and that is the right hand side that we were waiting for and hence we have proved the third part also and this ends the session. I hope you enjoyed just start from any of the side and take care that you need to reach on the other side and just focus on the area of the triangle that you need. Bye for now.