 Hello friends and welcome to this session on geometry we are dealing with circles and here is a problem which we are going to solve in this session what does the problem say friends it says in the given figure ab is parallel to cd okay so here ab you can see this is ab and this is cd they are parallel o f is perpendicular to cd right this is given here and this is 90 degree you can see it is already shown in the figure oe is perpendicular to ab where is oe this is oe this is perpendicular to ab here it is shown here very good and it is cd is 24 centimeter it is mentioned here cd is 24 centimeter ab is 10 centimeter so ab is 10 centimeter it is mentioned here and e f is 70 it is mentioned here right we have to find the length of the radius of the circle okay now let us clear these marks okay now the figure is self-explanatory here so how to approach this problem clearly you can see there are triangles that to right triangle there are 90 degree angles so hence what comes to your mind first application of Pythagoras theorem is it so it is a straightforward case of Pythagoras theorem application so let us do that and some information anyways have been given to you so let us utilize that now ab is parallel to cd it is given okay and o f is perpendicular to cd so we know that if o f is perpendicular to cd first let us start with writing what is given it is a mandatory thing to do because that gives the clarity of understanding to the person who is solving as well as if you are in an exam and if you want to display your knowledge of this particular topic then the examiner who is evaluating your script is going to see whether you have mentioned the right conditions right so o f is perpendicular to cd also o e is just a repetition of this thing but also keep checking whether whatever your writing is correct or not and cd is 24 cd is 24 centimeter ab is 10 centimeter it is given 10 centimeter okay and e f is 17 centimeter very good right right and what is the objective to find keep a command on the demand of the question to find radius of the given right so let us try and solve this question solution okay so we know that since since a perpendicular perpendicular from the center from the center bisects the chord correct so if you drop up a perpendicular from the center on to any chord it will bisect the chord so we know this is it so bisect bisect the chord okay so that means you can very clearly say cf is equal to fd and that will be equal to half of cd and that will be equal to what is cd guys so if you look at the given condition cd is 24 just verify yes 24 so cd is 24 so 24 upon 2 that is 12 centimeter no doubt so let us write that number here so cf is 12 okay also by the same logic you will say ae is equal to eb is equal to half of ab is that same logic the perpendicular drop from the center bisects the chord so half ab and ab is how much just check 10 centimeter right 10 centimeter expansion so hence it is 10 upon 2 hence 5 centimeter for better clarity let us write that value here fair enough now it's given that ef is 17 centimeter right so we will start with this let us say we will say that you know this part you call it x oe let's say oe is x so what is this 17 minus x is it 17 minus x because the distance ef this whole distance here if you see this this distance ef between the two parallel lines it is 17 centimeter so if you take one as x the other will be 17 minus x now since cd is parallel to ab the of and oe of this one this one of and oer collinear they are in the same line right since cd and ab are sorry parallel lines so if you drop a perpendicular from the same point on two parallel lines it will be falling on the same line so f oe are collinear okay so hence we can say x and 17 minus x if these lines were not parallel that is ab was not parallel to let's say ab was not parallel to cd then this logic this logic would not work right then the distance between them is not 17 that you should keep in mind okay now once we know this we can always write or take the help of Pythagoras theorem so you can say in triangle in triangle ofc what can I write I can write oc square is equal to of square plus fc square without doubt let it be number one and similarly let me now write it here okay so in triangle in triangle which one um oae look at oae look at the figure you will see oa square oa square is equal to oe square plus ea square okay and both these are because of you can write Pythagoras theorem Pythagoras theorem right without doubt very good so hence now since both are being equated to the radius so since oc is equal to oa is equal to r isn't it oc is equal to oa is equal to r so what we can do we can equate so from 1 and 2 you can say from 1 and 2 what can we say we can equate both these so you can say of square plus fc square is equal to um oe square plus ea square isn't it right so what is of guys of is 17 minus x so 17 minus x whole squared plus fc what is fc guys 12 square will be equal to oe square what is oe x look at the figure it is x we have mentioned it as x and ea square what is ea ea is 5 square that we have calculated already so let's proceed let's proceed further so what do we do use the identity a minus b whole square open this is brackets so 17 square minus 2 times 17 times x plus x squared plus 144 12 square is 144 okay and that will be equal to that is going to be equal to how much x squared plus 35 isn't it now if you see this x squared and this x squared can be cancelled okay because they are on two sides of the equation now uh two 17 square is let's simplify this this is 289 plus 144 isn't it minus 25 if you take this 25 on the LHS take this 25 in the LHS you'll get minus 25 and this is 34 x which will come on the RHS okay now let us make another column and try solving here okay now so what do we get 289 minus 25 which will be if you see um 4 and 6 264 right 264 plus 144 so let me write it here plus it will be easier to add so how much it is 8 4 plus 4 and now this is equal to 34 x isn't it so 4 plus 4 8 6 plus 4 10 4 408 is equal to 34 x so what is x my friends 408 by 34 okay this is and if you see this is 17 2 ja and 17 uh twice and you have 68 which is 4 so you'll get x as 12 centimeter fair enough so my dear friends x is 12 centimeter if x is 12 centimeter then my job is done so now I have to find out radius so ac or a o square is equal to x square plus 5 square which is let me just draw a margin here yeah a o square is equal to so a o will be under root x square how much 12 square plus 5 square and you know 12 and 5 are part of the pathogen triplet so hence a o is 13 centimeter okay so hence we could solve this problem and figure out the value of radius a o right so r is equal to 13 centimeter this is our finding so what is the learning in this question we used the property one theorem that a perpendicular from the center by 60 chord this was the important theorem so you must know this as well as the application of Pythagoras theorem as well as how to equate to given conditions and find out the right answer so that was that is what is learned in this problem so let's take another problem and understand that as well