 Hello and welcome to the session. In this session we will first discuss angle subtended by a chord at a point. Consider this chord PQ of the circle with center O. This angle that is the angle PRQ is the angle subtended by the chord PQ at a point R on the circle. As you can see that the point R is on the natured arc of the circle. Now we have a result which says that equal chords of a circle subtend equal angles at the center. Suppose that we are given that the chord PQ is equal to the chord RS. Now chord PQ subtends angle POQ at the center and chord RS subtends angle ROS at the center. So according to this result we have that angle POQ is equal to angle ROS. Since the chords PQ and RS are equal. Now then we have if the angles subtended by the chords of a circle at the center are equal then the chords are equal. Like in this case if we are given that angle POQ is equal to the angle ROS then we say that chord PQ is equal to the chord RS. Next we discuss perpendicular from the center to a chord. We have the perpendicular from the center of a circle to a chord bisects the chord. Like here we have we have drawn a perpendicular from O on the chord PQ then PM would be equal to MQ that is perpendicular from the center of a circle to the chord PQ bisects the chord PQ. Now the converse of this result is given by the line drawn through the center of a circle to bisect a chord perpendicular to the chord. In this case we have drawn a line through the center O of the circle such that it bisects the chord PQ that is we have PM is equal to MQ then we say that this OM is perpendicular to the chord PQ. Suppose that we are given that the radius of the circle is 13 centimeters that is we have OP is equal to 13 centimeters and length of one of its chords is 10 centimeters so we take PQ equal to 10 centimeters we need to find the distance of the chords on the center for this we draw OM perpendicular to PQ. So as you can see we have taken OM perpendicular to PQ and we know that the perpendicular from the center of the circle to the chord bisects the chord. So in this case we have PM would be equal to MQ and since we have PQ is 10 centimeters so PM equal to MQ would be equal to 10 upon 2 equal to 5 centimeters. Let's consider this right triangle OPM in this we have OP square would be equal to OM square plus MP square that is by the Pythagoras theorem. So now OP is 13 centimeters so 13 square is equal to OM square plus MP is 5 so 5 square that is we have 169 minus 25 is equal to OM square so from here we have OM square is equal to 144 which gives us OM equal to 12 centimeters that is distance of the chord from the center is 12 centimeters. Now we should discuss circle through three points we say that there is one and only one circle passing through three given non-colonial points. Suppose that we are given three non-colonial points A, B and C we see that there is only one circle which passes through three non-colonial points A, B and C. Consider this triangle ABC there is a unique circle passing through the three vertices A, B and C of this triangle that is the circle. This circle is called the circum circle of triangle ABC and its center that is the center O of the circle is called the circum center of triangle ABC and the circum radius of triangle ABC is the radius of the given circle like in this case if we take OB then we say that OB is the circum radius of triangle ABC. Now we should discuss equal chords and the distances from the center. We know that the length of the perpendicular from a point to a line is the distance of the line from the point that is length of the perpendicular OM is the distance of the line AB from the point O. So for this chord AB of the circle is the distance of the line AB from the point O. So for the distance of the circle its distance from the center O of the circle would be the length of the perpendicular OM from the center and we have a result which says that equal chords of a circle or of congruent circles are equidistant from the center or centers like if we have that the chord AB and chord PQ are equal then their distances from the center are equal that is their equidistant from the center that is we get that OM would be equal to OM where OM is also perpendicular from O to the chord PQ. Like if we have that the chord AB of the circle is the distance of the line AB from the point O of the circle is the equal to the chord PQ then their distances from the center are equal that is their equidistant from the center that is we have OM is equal to OM and the converse of this statement is given by chords equidistant from the center of a circle are equal in length that is if we are given that OM is equal to OM then we say that the chord AB is equal to the chord PQ. So this completes the session hope you have understood the angles subtended by a chord at a point the perpendicular distance from the center to a chord then circle through three given points and equal to the length of the chord PQ.