 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says AC and BD are quads of a circle which bisect each other. Prove that first AC and BD are diameters and second that A, B, C, D is a rectangle. So let us see what is given to us in the question. We are given a circle then we are given two quads AC and BD such that they bisect each other. That means AO is equal to OC and OB is equal to OD. Now we have to prove that AC and BD they are the diameters of this circle. So we start with the solution. You see the solution to the first part. First of all we join AB, BC, CD and AD. Now we see that in triangle AOB and triangle COD. That means this triangle and this triangle OA is equal to OC that we have seen because AC and BD they bisect each other. So this is equal to this. Secondly we see that OB is equal to OD and we see that the angle enclosed between these two sides that means angle AOB is equal to angle COD because these are vertically opposite angles. So vertically opposite angles are equal. So we see that by side angle site criterion triangle AOB is congruent to triangle COD. These two triangles are congruent therefore AB is equal to CD and this we get by CPCT that means corresponding parts of congruent triangles and we know that they are also equal. C is equal to CD therefore arc AB is equal to arc CD is equal to arc CD angles AOD and BOC is equal to arc VC. We name this one and this two arc AD is equal to arc CD plus arc VC that means is equal to arc VC plus arc CD or we can say that arc VAD divides or bisects that VD diameter. So by this if AB is equal to angle ABC is equal to angle VCD is equal to angle CDA and each of them is equal to 90 degree. So if we prove this and we can easily prove that ABCD is a rectangle. So let us start with the proof of congruent triangle that triangle BOC is congruent to triangle AOD. So for the same reason we will have that angle AO is equal to angle OCB. So similarly such that the pair of alternate angles are equal therefore these two lines will be parallel to each other. We can say that AB is parallel to CD and VC is parallel to have this that AB is parallel and VC is parallel to AD. This implies that AO therefore angle DAB the opposite angles this sum up to 180 degree therefore its opposite angles will be equal. So angle DAB is equal to angle BCD is equal to half of 180 degree. Angle DAB is equal to angle BC by 2 and that is equal to 90 degree. So we have proved that angle DAB is equal to angle BCD is equal to 90 degree. Similarly we can prove that angle CBA is equal to angle ADC is equal to 90 degree and enjoy the session. Have a good day.