 Hello and welcome to the session. In this session we discussed the following question which says, in the given figure, ABC is a right triangle, right angle at A, find the area of shaded region, if AB is equal to 6 centimeters, BC is equal to 10 centimeters and O is the center of the in-circle of triangle ABC. Before we move on to the solution, let's recall one result which says that the lengths of the two tangents drawn from an external point to a circle are equal. This is the key idea that we use for this question. Let's proceed with the solution now. This is the given figure in which we have triangle ABC is the right triangle which is right angle at A that is angle BAC is equal to 90 degrees, then we have AB is equal to 6 centimeters, BC is equal to 10 centimeters and we are supposed to find the area of the shaded region. First of all we consider the right triangle BAC in this we have BC square is equal to AD square plus AC square using the type of Gorus theorem. Now we substitute the values for BC and AB so we get 10 square is equal to 6 square plus AC square. From here we get AC square is equal to 10 square minus 6 square that is we have AC square equal to 100 minus 36 which gives us AC square equal to 64 and from here we get AC equal to square of 64 thus AC is equal to 8 centimeters. Now we won't be taking the negative sign because sign cannot be negative so we get AC equal to 8 centimeters. Now next we draw OD perpendicular to AB, OE perpendicular to AC is perpendicular to BC. So we have this OD is perpendicular to AB, OE is perpendicular to AC and OF is perpendicular to BC. We take net R be the radius of the encircle that is we have OD is equal to OE is equal to R. Now in the figure OD on angles R of measure 90 degrees since it is given that this angle is 90 degrees now we have drawn perpendicular so this is 90 degrees and this is 90 degrees and obviously this also would be 90 degrees. So all the angles are of measure 90 degrees and the adjacent sides OD and OE are equal to R. Therefore we can say that the figure AE OD is a square. Now since it is a square so all its sides would be equal and thus we get OD is equal to OE is equal to AD is equal to AE each equal to R. But since we got AC equal to 8 centimeters therefore from the figure you can see that CE is equal to AC minus AE thus we get CE is equal to 8 minus AE that is R. So 8 minus R centimeters is CE now from the figure BD is equal to AB minus AD thus BD is equal to AB which is 6 centimeters minus AD that is 6 minus R centimeters is BD. Now CF would be equal to CE since these are the tangents from the point C to the circle and therefore they would be equal. Now since we had CE equal to 8 minus R centimeters thus we get CF equal to CE equal to 8 minus R centimeters. In the same way we have BD is equal to BS as these are the tangents drawn from the external point B to the circle. Now BD is equal to 6 minus R centimeters therefore we get BD is equal to BS and each is equal to 6 minus R centimeters. We are given that BC is equal to 10 centimeters so from the figure we have BS plus FC is equal to 10 centimeters BS is 6 minus R plus FC or CF which is 8 minus R this is equal to 10 thus we get 14 minus 2R is equal to 10 so from here we get 2R is equal to 4 so this gives us R is equal to 2 centimeters thus the radius of the N circle ABC is equal to 2 centimeters. We need to sign the area of the shaded region area of the shaded region is equal to area of triangle ABC minus area of the circle so this is equal to area of triangle ABC would be equal to half into base which is AB into height that is AC minus the area of the circle which would be pi R square and this R is the radius of the N circle of triangle ABC. Now putting the respective values we get this is equal to half into 6 into 3.14 into R square that is 2 square now this 2 3 times is 6 and so this is equal to 3 multiplied by 8 which is 24 minus 4 into 3.14 this is equal to 24 minus 12.56 which is equal to 11.44 centimeter square thus we get area of the shaded region is equal to 11.44 centimeter square this is our final answer so this completes the session hope you have understood the solution of this question.