 Hello and welcome to another problem-solving session. In the given question A0 by OC Okay, and B0 by OD is equal to half. So here is the first condition which is good and AB is equal to 5 centimeters. So let me write it here. So this is 5 centimeters and We have been asked to find out the value of BC this correct This is what we have to find out Okay, so how to go about it. So what's given? So given is A0 the ratio A0 upon OC is equal to B0 upon OD is equal to half This ratio has been given and what do I what do we know? If in any given quadrilateral, so this is a quadrilateral ABCD and the diagonals divide The angles are dividing themselves Proportionally, so then we know that the given quadrilateral will be a trapezium, right? So we'll write since in a Alright, if instead of writing since you'll write if if in a given Quadrilateral Quadrilateral We're gonna give in quadrilateral diagonals So clearly here diagonals are AC and BD diagonals divide Divide each other each other Proportionally Proportionally, right? Then what is the conclusion then? The quadrilateral Quadrilateral is a Trapezium right this is a established fact so it gives us a lot of help the moment we established that ABCD is a trapezium therefore ABCD is a trapezium is a Trapezium Trapezium and now you know which two sides will be parallel so AB is parallel to CD AB is parallel to CD Right, so you can argue why would AD not parallel to BC But AD would not be parallel to BC because if AD were to be parallel to BC then the ratio would be Then the equal ratio should have been AO by OC is equal to BO by OB But that's not the case is the other way around so hence this is not not true So hence a AD is not parallel to BC AB is parallel to BC, right? So we'll put the parallel symbol here Okay, so these are these two lines are parallel the moment. I say these two lines are parallel Don't we see that this will become equal to X So this is X alternate interior angle and if this is Y Then this becomes one correct. So now let me just remove these things. These are not important Now let me write this in triangle. So I'm writing here from here in Triangle a OB or Yeah, a OB and Triangle DOC DOC Okay, DOC DOC what do we know, you know that angle OAB is equal to angle OCD Right, you just do that. This is alternate interior angles and Similarly angle ABO Is equal to angle CDO CDO Right again because of the same thing. So therefore I can say that Triangle AOB is similar to triangle COD COD by SS or sorry by a criteria, correct? Now that is so then we can clearly say that AO by OC will be equal to AB by CD Right, so clearly we can find out CD is equal to AO by OC Into a CD CD is equal to sorry. This will be the other way around This will be OC by AO into AB Now what is OC guys OC? sorry OC by OC by AO is 2 so this ratio already I know this is 2 Into AB AB is 5 so hence I'm sorry alternatively Alternatively the same question could have been solved in a very easy or a very quick manner and this is This is how you have to do it. So given is AO by OC is equal to BO by OD so AO by OC is equal to BO by OD Isn't it and and Angle AOB is equal to angle angle COD COD therefore by SAS similarity criterion similarity criterion criterion triangle ABC or AOB is similar to triangle COB Correct. So this was our one-liner similarity proof. So we could have avoided the trapezium route, but anyways So if there are multiple ways of solving the same problem So hence what will happen and why is this angle by the way equal because of vertically opposite angles? You can see that therefore I can I can say now AO by OC will be equal to AB by CD again and again you will find out that CD is equal to OC CD is equal to sorry CD is equal to AB into OC by OA Which is 5 into 2 by 1 hence 10 centimeters. So this is Another way of doing it. In fact, this is much simpler way than the previous method But the same problem can be established or proven through multiple routes as well in similarity