 Welcome to this question of the day problem. In this question, it's given that the circle is inscribed in a 3, 4, 5 triangle. That means the dimensions of the sides are 3, 4 and 5. As shown in the diagram, what is the radius of the circle? So we have to find out what is the radius of this circle. So if you see this circle is the in circle inscribed within the triangle. So we have to find out the radius of that circle. Now clearly if the sides are 3, 4, 5, then by the converse of Pythagoras theorem, this triangle happens to be a right angle triangle and here is the 90 degree angle. Now how to find out the radius? So the theorem which we are going to use is that the tangents drawn from a point onto a circle are equal. So if you see, I have drawn this diagram here. Now in this triangle A, B, C, let us say this point O is the center of the circle. This is point O, this one now and we see is what we see is A, B, C and A, C are tangents to the circle, isn't it? Because what is a tangent? Tangent is the line which is touching the circle exactly at one point. So A, B, C and A, C are the tangents. Simultaneously we can also say that BD, BD, where is BD? So if I have to highlight, this is VD, VD and FB are the tangent to the circle, right? Now BD and BFR from the same point B, so we can say that both of them are of the same length and I have shown that as X here. Similarly, CD is equal to CE, so I am writing here like this, BD is equal to BF, why? Because of this theorem and similarly CD is equal to CE and BD is equal to BF, I have assumed it to be X, so automatically CD will become 3 minus X, why? Because this side BC was of 3, so this is 3 units, so this is how the configuration would look like. Similarly, AE is equal to AF and that is equal to 4 minus X, isn't it? So you can actually see that 4 minus X or in order this is also equal to 5 minus 3 minus X which is equal to 2 plus X, how am I writing all this? So if this is X, this happens to be 3 minus X and if this is X, this happens to be 4 minus X and now this is 3 minus X, CE is equal to, so this CE is equal to 3 minus X, why? Because this is given theorem and similarly AE will be equal to AF, AE this side will be equal to this side, AE is equal to AF and that will be equal to 5, total length was 5 and 5 minus 3 minus X is equal to 2 plus X, I hope you understood these. Now clearly from these two relationships, I can say that 4 minus X that is, what is this? This is AB is equal to 2 plus X, what is this? This is AE, I am sorry this is not AB, this one is AF, this one is AF and this one is equal to AE, isn't it? So if you solve these you will get 2X is equal to 2, so X equals to 1, isn't it? This is one method of doing it, another method of doing this could be, you know that area of triangle ABC is equal to area of triangle, let us see what all triangles it has made up of AOF, triangle AOF plus area of triangle AOE, I believe yes AOE, AOE plus area of triangle BOF, triangle BOF plus area of triangle BOD, triangle BOD, yes triangle BOD and plus area of triangle COE plus area of triangle COD. The total area is nothing but some of all those small triangles and if you see, what will be the area? So if you see area of triangle ABC is half into base into height, so you can get half into 4, sorry this is not 1 by 3, this is 1 by 2, so you can get half into base, this 3 into height is this thing. Now in area triangle AOF, you see what is this, triangle AOF is this, this is nothing but half into 4 minus x into r, is it it?