 Hello and welcome to another session on circles and As we have been doing so far. We are trying to validate each and every theorem what we are Proving in the second part. So here is another theorem. It says line through the centers of two intersecting circles Bisex the common chord perpendicularly. That means if you see there are two circles with center O and O dash AB is the common chord A and B are the points of intersection of the two circles AB is the common chord and O dash is the line segment joining the two circles and We have to prove that this line segment O dash by six AB perpendicularly that is OCA or OCB is 90 degrees and CA is equal to CB Okay, so before we Get in get on to the proof. Let's first see whether that is actually true. So Let's first try and measure some lengths. So if you see Measuring AC Okay, so 2.14 Okay, so let me just do a little bit of setting change if you see Um Yeah, so I'm just removing the background color. So then come easier for you to see so AC is 2.14. Let us also measure CB so or BC. So this is also 2.14, right? So same way. Let us just change the settings and color Now so This is 2.14. Okay, now we will try to see different configurations and see if it is true always So here I am going to change the circle positions C so I'm changing the circle so the two intersecting circles Now intersecting at different different points Can you see AB the location of A and B is changing and every time A and B are the positions are changing You can see AC and CB stays the same. There is no difference whatsoever. That means it's always the same. Okay So this was in one case. Let us say if we are changing the circle position itself Yeah, so now I'm changing it to that's it Like that. So here also if you see CB and AC are same Okay Right the same every time doesn't matter which Configuration is see now. I'm changing the first circle So whatever it is in every such case you will find that oh dash the line joining the two centers always divide or bisect B to bisect the common chord and not only bisect it's also Biscecting it perpendicularly, right? So this is what you wanted to demonstrate that, you know any such you know Two circles which when they intersect the common chord gets bisected by the Line joining the two centers Okay, so whether the centers are within one circle or they are outside of each other. Okay, so this is Established now what we are going to do is see even I you know when the Distance is too large. There is no common chord. So hence no point Yeah discussing that so here is the common thing and Yes, so hence we are going to prove this now friends so we just saw the Validation of this particular theorem that line through the centers of two intersecting circle Circles bisects the common chord perpendicularly. We just saw this on geojibra and we tried to Get different configurations of the two circles and every time this particular theorem was valid, right? Now we are going to prove this So we have already mentioned what's given so a and b are points of intersection of the two circles The two circles are mentioned over here. We have to prove that oh dash is perpendicular to ab and ac is equal to bc That's what we need to prove. So let's start with The proof so we are going to consider What what is the you know? The direction for this proof so We know that ac is equal to bc can be proven if Ac and bcr the corresponding parts of two congruent triangles, isn't it? So but which two triangles if you see closely Oac and obc are the two triangles if you prove somehow to be congruent then ac automatically will be equal to bc Isn't it? So let's try to prove that these two congruence these two triangle to be congruent But then for that we need few more information So given information is oa is equal to ob why because they are the radii. So this is true So oa is equal to ob this is okay oc is also a common side But then we have only these two information nothing else Correct. So if I somehow prove that this angle x is equal to this angle y Then my job is done, right? So for proving this we need to prove that x and y are to be equal or any other angle maybe this angle Yeah, so included angle only because two sides are equal So I have to somehow prove that x equals to y or somehow prove that c is 90 degree But proving x equals to y would be much easier. Why because if you consider these two triangles So which two triangles guys so triangle a o o dash and Triangle b o o dash Okay, so in these two triangles I have o a is equal to ob Right radii of same circle. They are radii of same Circle, isn't it? similarly o dash a is equal to o dash b same logic radii of radii of same circle, right and o o dash is equal to o dash o common common side, isn't it? common side therefore we can conclude that Triangle a o o dash is congruent to triangle b o o dash Is it? So if that is true guys, then can't we say that angle a o o dash is equal to angle b o o dash so we can say a o o dash is equal to b o o dash y Corresponding parts of con congruent triangles, right now. This is what we wanted to achieve, isn't it? So hence now let's consider triangle a o C and Triangle b o c so we have in this O a is equal to ob radii of the same circle radii radii of same Circle Isn't it then? angle a o c is equal to angle b o c just proved just proved here Right, though it is saying a o o dash, but a o o dash is a o c and we have also Oc is equal to o c common side Common right therefore by what law or what criteria these two triangles are congruent so by by s a s criteria By s a s criteria by the way, this was triple s criteria, right? Side side side it will be side angle side in this case triangle a o c is congruent to triangle B o c Right, so therefore we can conclude that a c is equal to bc First we prove that c is the midpoint What else also? Also angle a co is equal to angle bco Right cpct both are because of cpct but we know that angle a c o plus angle bco is equal to 180 degrees Right guys, so hence angle a co Twice angle a co will be 180 degrees why because a co is equal to bco so hence angle a co is equal to 90 degrees Right hence proved both the things are proved hence proved so o o dash Not only bisects the Card a b why do we know and how do we know that o o dash bisects because a c is equal to bc We just proved so c is the midpoint not only c is the midpoint But angle a co is 90 degree that this is a second part of the proof, right? So hence we could prove that the line joining the centers of two intersecting circles Bisects the common chord perpendicularly in the first part We evaluated through the geojibra software in the second part. We proved it