Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
Prove exactly as Theorem 9.1 by considering chords of congruent circles.
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Use SAS axiom of congruence to show the congruence of the two triangles.
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.
6 cm. First show that the line joining centres is perpendicular to the radius of the smaller circle and then that common chord is the diameter of the smaller circle.
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.
If AB, CD are equal chords of a circle with centre O intersecting at E, draw perpendiculars OM on AB and ON on CD and join OE. Show that right triangles OME and ONE are congruent.
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.
Proceed as in Example 2.
If a line intersects two concentric circles with centre O at A, B, C and D, prove that AB = CD.
Draw perpendicular OM on AD.
Three girls Reshma, Salma and Mandip are standing on a circle of radius 5 m. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6 m each, what is the distance between Reshma and Mandip?
Let KR = x m. Area of ΔORS = 1/2 × x × 5. Also, area of ΔORS = 1/2 × RS × OL = 1/2 × 6 × 4. Find x and hence RM.
A circular park of radius 20 m has three boys Ankur, Syed and David sitting at equal distances on its boundary, each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.
Use the properties of an equilateral triangle and also Pythagoras Theorem.
In Fig. 9.23, A, B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.
45°
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.
150°, 30°
In Fig. 9.24, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.
10°
In Fig. 9.25, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.
80°
In Fig. 9.26, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.
110°
ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.
∠BCD = 80° and ∠ECD = 50°
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
Draw perpendiculars AM and BN on CD (AB ∥ CD and AB < CD). Show ΔAMD ≅ ΔBNC. This gives ∠C = ∠D and, therefore, ∠A + ∠C = 180°.