 Hi friends, so we are going to discuss another very very important Theorem in circles this particular theorem is going to help you prove many other theorems As well as you will be able to solve many problems related to circles and its properties So let us understand this theorem. It says the perpendicular from the center of a circle to a chord Bisects the chord, right? You understand what a perpendicular is. So you are dropping a perpendicular from the center O in this case on to a chord PQ and The foot of the perpendicular is R as shown in the picture. You have to prove That PR is equal to RQ. That's what is this word bisect means, okay Now let's Prove it. So what's given? It is given that O R is perpendicular to PQ right right and O is the center O is the center of Circle Okay, you have to prove to prove That PR is equal to RQ. Then we will say that O R bisects the chord okay, how to prove proof is very straightforward and We know that we have to use something called congruence of triangles which we have studied in this grid so We have to basically prove that PR is equal to RQ. So whenever There are two items two geometrical elements to be proven to be equal We have one methodology and that is, you know, if we somehow prove that they are corresponding parts of Two congruent triangles, then we are done. So It looks like they are part of two congruent triangles. So let's see what all so if you look closely triangle O PR so you can write in triangle O PR and triangle O Q R I have Maintained the order of the vertices. So you have to be very very careful about it. So O PR and O Q R So clearly we see we have to find two triangles two different triangles PR should be part of one triangle RQ should be part of the other triangle then and if we somehow prove that PR and RQ are the corresponding parts of those two triangles and You know, those two triangles are also congruent then our job is done So it looks like yes, it is like that only so hence let's evaluate these two triangles So in turn in triangle O PR and O Q R What is very obvious is O P is equal to O Q both are a radii radii of the same circle of The same circle, isn't it radius of the same circle Radio of the same circle are always equal. That's the virtue of the circle O P is equal to O Q now O R is equal to O R common This is a common side to both the Both the triangles, okay, and angle O O R P is equal to angle O R Q and both are equal to 90 degrees, right? Why because we know that O R is perpendicular to P Q This was given here see right So hence by which property guys if you see there's a 90 degrees there is a hypotenuse This R is the hypotenuse of both And other side so hence we say triangle O PR Is congruent to triangle O Q R again order of the vertices very very important, okay therefore We will write PR is equal to Q R Is it it why this is C P C P, right? And that's what we wanted to prove so PR is equal to Q R hence Proved so what do we learn out of this theorem that a perpendicular From the center dropped on to a chord not just any perpendicular the perpendicular has to be dropped from the center So if you drop another perpendicular like that that will not bisect the chord P Q Okay, only that chord which is only that perpendicular which is dropped from the center on to the chord Will bisect the chord, okay, and we just proved this in fact if you see the converse is also true Which will prove in the next session So the converse of this theorem will be that if you join the midpoint of a chord to the center That that line will be perpendicular to the P Q Or to the chord, isn't it once again if you join the midpoint So, you know we prove that R is the midpoint of P Q But let's say R is given to be the midpoint of P Q and you join it to the center Then you have to prove that the that line segment that is OR Is perpendicular to the P Q that will be the converse statement of this given theorem So we will study we will study that converse in the next session