 Hi and welcome to the session. I'm Kanika and I'm going to help you to solve the following question. The question says a chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc. Before solving this question, we should first be well versed with theorem 10.8 and theorem 10.11 given in your NCERT book. Theorem 10.8 states that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. This theorem means that if we have a circle in which arc A V subtends angle A O B at the center and angle ACB at point C on the remaining part of the circle, then angle A O B that is angle subtended by arc A V at the center is two times angle ACB that is angle subtended by the same arc A V at point C on the remaining part of the circle. You should also know that a quadrilateral A B C D is called cyclic if all the four vertices lie on a circle and theorem 10.11 states that the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree. This theorem means that angle A plus angle C is equal to 180 degree and angle B plus angle D is equal to 180 degree. The knowledge of these two theorems is the key ideas in this question. Let's now begin with the solution. Let's first make a diagram of this question. We have drawn a circle whose center is at O. Now in the question we are given that a chord of a circle is equal to the radius of the circle. So let A V be the chord of this circle and O is the radius of this circle. Now it is given to us that A V is equal to O A. So we are given that O A is equal to A B. Now in construction we will draw a line through O with joints O to B. So let's now join O B. Now O A is equal to O B because they are radius of the same circle and we are given that O A is equal to A B. Let's name this equation as equation number one and this as two. Now from one and two we have is equal to O B is equal to A B. O A is equal to O B is equal to A B. So this implies triangle O A B is an equilateral triangle and since triangle O A B is an equilateral triangle therefore each angle of this triangle is of 60 degree right and hence angle A O B is equal to 60 degree. Now look at the question again. We have to find the angle subtended by the chord that means by A B at a point on the minor arc and also at a point on the major arc. Let C be a point on the major arc A B. Let's now join C to A and C to B. So now we have to determine angle A C B. Now since A B subtends angle A O B at the center, angle A C B at point C on remaining part of the circle therefore angle A O B is equal to two times angle A C B by theorem 10.8 which we have learned in key idea and which states that the angle subtended by an arc the center is double the angle subtended by it at any point on the remaining part of the circle. Angle A O B is equal to 60 degree so we have 60 degree is equal to two times angle A C B this implies angle A C B is equal to 30 degree. So angle subtended by the chord A B at a point on the major arc is 30 degree. Now we will find the angle subtended by chord A B at a point D on minor arc A B. Let's now join A to D and B to D. By joining A to D and B to D we now have a quadrilateral A A D B C. Now as vertex B C and D of quadrilateral A B C D lie on a circle therefore A B C D is a cyclic quadrilateral and this means angle C plus angle D is equal to 180 degree because sum of opposite angles of a cyclic quadrilateral is 180 degree. We know that angle C is equal to 30 degree right. So 30 degree plus angle D is equal to 180 degree and this implies angle D is equal to 150 degree. So angle subtended by chord A B at point D in the minor arc A B is 150 degree. Hence our required answers are 150 degree and 30 degree. This completes the session. I and take care.