 Hello and welcome to the session I am Deepika here, let's discuss the question which says in the circle of radius 21 centimetre nR subtends an angle of 60 degree at the centre. Fine. 1. The length of the arc. 2. Area of the sector formed by the arc. 3. Area of the segment formed by the corresponding corp. Let us first understand how to find the length of the arc APB. Now we know that the circumference of the circle is 2 pi r that is the whole length of the circle of angle 360 degree is 2 pi r. So when degree measure of the angle at the centre is 360 then length of arc is 2 pi r. So when the degree measure at the centre is 1 then length of arc is 2 pi r upon 360. Therefore when the degree measure of the angle at the centre is theta then length of arc is 2 upon 360 upon 360 into theta. The formula for the length of the arc APB that is the length of sector angle theta equal to theta upon 360 into 2 pi r. Again we know the area of the sector of angle theta which is equal to theta upon 360 into pi r square. Now in this figure AB is a chord of the circle with centre O. So this portion APB is a segment of the circle. Now area of the segment APB is equal to APB minus area of triangle OAB. So this is a key idea behind our question. Take the help of this key idea to solve the above question. So let's start the solution part 1. In part 1 we have to find the length of the arc of a circle whose radius is 21 centimeter. The arc subtends an angle of 60 degree at the centre. So we are given AOB is equal to 60 degree radius is equal to 21 centimeter. According to our key idea the length of the arc is given by the formula of a sector angle theta is equal to theta upon 360 into 2 pi r. Now here r is 21 centimeter and theta is 60 degree. So this is equal to 60 upon 360 into 2 take pi is equal to 22 by 7 into r. So on cancellation we have the length of the arc is equal to 22 centimeter. Hence the answer for part 1 is 22 centimeter. Let's move to the part 2. In part 2 we have to find the area of the sector formed by the arc. Now we know the formula for area of the sector angle theta is equal to 2 upon 360 into pi r square. Now sector formed by the arc is equal to area of sector OAPB equal to 60 upon 360 into 22 upon 7 into 21 into 21 centimeter square. So on cancellation we have. So this is equal to 11 into 21 centimeter square or this is equal to 31 centimeter square. Hence the answer for part 2 is 231 centimeter square. Let's move to the part 3. Part 3 we have to find the area of the segment formed by the corresponding arc. So according to our key idea area of the segment APB is equal to sector AB minus area of triangle OAB. Now in part 2 we have found the area of sector OAPB which is equal to 231 centimeter square. Now we will find the area of triangle OAB. Let's draw the triangle separately perpendicular to AB. So for finding the area of triangle OAB draw perpendicular to AB. Triangles AM OBM we have A is equal to OB radii of the circle and M is equal to OM common. Therefore triangle OAM is congruent to triangle OBM by RHS congruence condition. M is the midpoint of AB and angle AOM is equal to angle BOM is equal to 30 degree. Now the formula for the area of triangle is half base into height that is we want to find AB as well as OM is equal to x centimeter from triangle OAM upon OAM is equal to cos 30 degree. Now OM is x centimeter so x upon OA which is 21 centimeter is equal to root 3 by 2. Therefore x is equal to 21 into root 3 by 2 centimeter. Now A is equal to sin 30 degree so AM upon 21 is equal to 1 by 2 as sin 30 degree is 1 by 2. Therefore AM is equal to 21 by 2 centimeter. Hence AB is equal to twice AM because AM is the midpoint of AB and this is equal to 2 into 21 by 2 centimeter or this is equal to 21 centimeter. So area of triangle OAB is equal to 2 base that is AB into height OM. Now this is equal to 1 by 2 into AB is 21 centimeter into OM. OM is 21 root 3 by 2 we have this is equal to 441 root 3 upon 4 centimeter square. Therefore the area of the segment AB is equal to factor which is 231 centimeter square we have proven part 2 minus area of triangle OAB this is 441 root 3 by 4 centimeter square hence the answer for this part is 231 minus 441 root 3 upon 4 centimeter square. I hope the solution is clear to you bye and take care.