 this question it's been given that in the given figure pq is parallel to mn so pq is parallel to mn these two lines are parallel and kp upon pm is given as 4 upon 13 so hence kp upon pm so kp is this upon pm is given as 4 upon 13 and kn value is given kn length total kn so this length has been given and that is equal to 20.4 centimeter you have to find the value of kq so let me highlight what we need to find out okay so how to go about it clearly the moment I see a triangle there is a triangle and then pq is parallel to mn the moment I see that I see that first thing which comes in my mind is basic proportionality theorem or Thales theorem and then they are also talking about the ratios and basic proportionality theorem also talks about ratios so hence probably this question requires the application of basic proportionality theorem or Thales theorem okay so we know so let's how do we start the solution so you have to write given okay so given is pq is parallel to mn so though it's a mundane work to repeat the same thing but it's always a good practice to you know record all the given facts and then you can also write kn is equal to 20.4 centimeter to find what is to be found out kq okay now solution so how to solve it so you can write by Thales theorem Thales theorem or by PPT basic proportionality theorem what do I know about this triangle so we can say a kp upon pm to be equal to kq upon qn isn't it this is what the basic proportionality theorem talks about so if pq is parallel to if one line is parallel to the third side of a triangle and the line cuts the two other sides at two points p and q then kp upon pm ratio is equal to kq upon qn that's what the basic proportionality theorem says so now the rest everything is just plain and simple mathematics mathematical manipulation so you can what you do is you can take invertindo or you the reciprocal of this will be same so pm by kp pm by kp is equal to qn by kq why did I do this you will now come to know because we have to find out see kn is given is it guys kn is given or alternatively what you can see is say let us say this is equal to x so this is 4x and this is 13x okay so 4x and 13x so let us say we have to find out kq right so let us say kq is y so clearly qn is 20.4 minus y isn't it so you can write it here and then and then what you can do is you simply apply basic proportionality theorem so if you see what will this result be so we can say 4x upon 13x is equal to y upon 20.4 minus y that's what this will result to right so xx gets cancelled so it is simple linear equation in y so it is 4 times 20.4 minus 4 y so I'm doing cross multiplication is 13 y so this thing goes and multiplies with this this thing goes and multiplies this with this whole is that fine so hence what is the result result is 17 y is equal to 81.6 right so 4 into 20.4 is 81.6 and then I took this minus 4 y and took it here and added to 13 y so you will get this 17 y is equal to this so what is y 81.6 upon 17 which comes out to be 4.8 centimeters okay so we found out y and that's what we have to find out we had to find out the length of kq so hence you can write kq is nothing but y is equal to 4.8 centimeter and this is the result okay so what did we learn in this problem in this problem so we learned the application of basic proportionality theorem and rest was the game of finding one value after one value if two proportions are given correct