Cord, secant relation while two circles touch externally

Hello maths bugs(🐞) and hivers(🐝)
I hope you are strong and stout & doing good in life.

Well come to another interesting circle ,cord and secant related problem.You have already seen the problem figure. First give it a try yourself and then if problem comes, go ahead to the solution and proof section below.

Let me show you the the ratios(equation) that will help us to solve the problem very easily. Check the problem figure once more and it will be given by as follows:

        AO×DO = BO×CO 
        Or, AO =BO×CO/DO 
        Or, AO =12×8/10 
        Or, AO =9.6 (unit)

LET'S PROVE IT

Drawing: We need to draw a common tangent of the two circle at point O..We also need to connect AB and CD with straight lines as shown in following figure:

You must know the ∆ABO and ∆CDO are similiar triangle.

Proof: From ∆ ABO and ∆CDO , we have

1️⃣ ∠AOB = ∠COD (Vertically opposite angle)
2️⃣ ∠ABO = ∠CDO (It is proved in the following figure)
3️⃣ ∠BAO = ∠DCO (When two angle equals, 3rd angle will also be equal)

WHY IS ∠AOP EQUALS TO ∠ABO ?
OR, WHY IS ∠COQ EQUALS TO ∠CDO ?
PROOF IS HERE.

So, we have ∆ABO ~ ∆CDO

I have already showed you how to find relation between sides of similiar triangle; I am not showing it here.

Now from similarity we can find AO×DO =BO×CO

How to use similarity very easily without any confusion, details is here.

There is another concept like this ; don't be confused while solving them. Check it below:

Check the difference between two :

I hope liked my today's problem and it's solution.

Thank you so much for stopping by.

Have a good day

All is well.

@Regards: @ meta007