Ex Fill in the blanks. Solution: i interior ii exterior iii diameter iv semicircle v the chord vi three. Write True or False. Give reason for your answers. Recall that two circles are congruent, if they have the same radii. Prove that, if chords of congruent circles subtend equal angles at their centres, then the chords are equal. Draw different pairs of circles. How many points does each pair have in common?
What is the maximum number of common points? Solution: Let us draw different pairs of circles as shown below: We have. Suppose you are given a circle. Give a construction to find its centre.
Solution: Steps of construction : Step I : Take any three points on the given circle. Let these points be A, B and C. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord. Let PQ be the common chord. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
Join OE. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords. Solution: Given : Two circles with the common centre O. Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park.
If the distance between Reshma and Salma and between Salma and Mandip is 6 m each, what is the distance between Reshma and Mandip?
A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone. Let the length of each side of the equilateral triangle be 2x. 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.
Solution: We have a circle having a chord AB equal to radius of the circle. Solution: The angle subtended by an arc of a circle at its centre is twice the angle subtended by the same arc at a point pn the circumference. In figure, A, B and C are four points on a circle. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. Solution: Since angles in the same segment of a circle are equal.
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. If the non � parallel sides of a trapezium are equal, prove that it is cyclic. Two circles intersect at two points B and C. Solution: Since, angles in the same segment of a circle are equal. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
They intersect at a point D, other than A. Let us join A and D. Thus, D lies on BC. Case � I: If both the triangles are in the same semi-circle. Join BD. DC is a chord. Case � II : If both the triangles are not in the same semi-circle. Prove that a cyclic parallelogram is a rectangle.
Since, ABCD is a cyclic quadrilateral. Thus, ABCD is a rectangle. Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.
Two chords AB and CD of lengths 5 cm and 11 cm, respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.
Solution: We have a circle with centre O. Let r cm be the radius of the circle. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre? Parallel chords AB and CD are such that the smaller chord is 4 cm away from the centre. Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle.
Proof: An exterior angle of a triangle is equal to the sum of interior opposite angles. Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.
Taking AB as diameter, a circle is drawn. A circle drawn with Q as centre, will pass through A, B and O. ABCD is a parallelogram. ABCE is a cyclic quadrilateral. AC and BD are chords of a circle which bisect each other. Similarly, AC is a diameter. Since, opposite angles of a parallelogram are equal. Two congruent circles intersect each other at points A and B. Solution: We have two congruent circles such that they intersect each other at A and B.
A line segment passing through A, meets the circles at P and Q. Let us draw the common chord AB. Since angles subtended by equal chords in the congruent circles are equal. The perpendicular bisector of BC passes through O. Join OB and OC. Suppose it cuts circumcirde at P. Solution: i interior ii exterior iii diameter iv semicircle v the chord vi three Ex Solution: Let us draw different pairs of circles as shown below: We have Figure Maximum number of common points i nil ii one �i two Thus, two circles can have at the most two points in common.
RD Sharma Class 12 Solutions. Watch Youtube Videos.
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