 So, we discussed this in the previous session that we are going to discuss about the converse of the theorem that a line dropped as a perpendicular from center onto a chord will bisect the chord. That was the theorem which we discussed in the previous session. Now, in this session, we are going to discuss its converse and what is the converse? It says the line joining the center and the midpoint of a chord is perpendicular to the chord. Though I have shown it to be perpendicular already, but I will not be starting from there. So, what is meant by this theorem is R is the midpoint. So, let me start with the proof straight away because it is very simple proof. So, what is given? So, we will first write given, it is given that PQ is the chord, is the chord or is a chord of a circle, circle with center O and R is the midpoint, is the midpoint of chord PQ. PQ. We have to prove to prove. What do we need to prove? We need to prove that O, R is perpendicular to PQ. So, let us see how do we prove it? So, the moment something has to be proven to be perpendicular, what comes to our mind? Either we go through Pythagoras theorem route or if we prove that the two angles are part of two congruent triangles and also linear pair, then also we can prove 90 degrees. So, I think the second case will work over here. Let us try to build on from there. So, consider triangle. So, in triangle O, P, R and triangle O, Q, R. What do we have guys? We have O, P is equal to O, Q, both are equal to R radius, radius, radius, radii of the same circle, same circle and then what next O, R is equal to O, R common side. This is a common side to both the given triangles. What else? Since R is the midpoint, so can I say P, R is equal to Q, R. Since R is the midpoint given, midpoint of PQ, right, therefore what do we infer from this? We infer that triangle O, P, R is my dear friends congruent to triangle O, Q, R, right. The moment we could prove that, we can say angle O, R, P is equal to angle O, R, Q and why is that? C, P, C, P, okay. So guys, if you see angle O, R, P plus angle O, R, Q is 180 degrees Y. We can say clearly this is linear pair because it was given that PQ is a straight line that is a chord, so hence it is a linear pair. So O, R, P and O, R, Q are same from here so we can say twice angle O, R, P is 180 degrees. So hence O, R, P is 180 degrees upon 2 which is 90 degrees, right. So the moment we got O, R, P as 90 degrees, we can say O, R is perpendicular to PQ, right. This is what we needed to prove, okay. So O, R was angle O, R, P is 90 so O, R is perpendicular to P, R and P, R is a part of PQ so O, R is perpendicular to PQ, hence proved quadrat Raman's Tendham.