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### High School Mathematics - 28.7 Circles

 Radius: A line joining the centre to any point on the circle. Diameter: Two times the radius is the diameter. Chord: A line segment joining any two points on the circle is chord. Major Arc and Minor Arc: Any two points A and B of a circle divide the circle into two points. If the two parts are unequal, the smaller part is called minor arc and the larger part is called major arc. A circle divides the plane into three parts, its interior exterior and the circle itself. The circle along with its interior is called the circular region. A chord divides a circular region into two parts called the segments of the region. The smaller segment is called the minor segment and the larger segment is called the major segment Theorem 1: In a circle, the perpendicular from the center to the chord bisects the chord. Theorem 2: Chords of a circle which are equidistant from the centre are equal. Given: OM = ON (radii of the circle) To prove: AB = CD Construction: Join AO and CO Proof: Consider triangles AMO and CNO OM = ON Angle M = Angle N = 90o Angle AOM = CON (common angle) Hence triangles AMO and CNO are similar. Hence AB = CD Directions: Solve the following.
 Q 1: In a circle of radius 5 cm, AB and AC are two chords such that AB = AC = 6cm, find the length of the chord BC.Answer: Q 2: AB and CD are two equal chords to a circle of centre O. If OM is perpendicular to AB and ON is perpendicular to CD. Prove that angle OMN = angle ONMAnswer: Q 3: Find the missing angle.Both angles are 40 degrees.Both are 50 degreesBoth angles are 35 degrees. Q 4: PQ and RS are two parallel chords of a circle whose radius is 10 cm. If PQ = 16 cm and RS = 12 cm, find the distance between the chords if they lie on the same side of the centre.Answer: Q 5: AB and CD are two parallel chords of a circle lying on the opposite side of the centre. AB = 10 cm and CD = 24 cm. If the distance between the chords is 17 cm, find the radius of the circle.Answer: Q 6: Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other on the same side of the centre. If the distance between the chords is 3 cm, find the radius of the circle.Answer: Q 7: Find the missing angle.100 degrees260 degrees150 degrees Q 8: Find the missing angle.50 degrees75 degrees110 degrees Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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