NCERT Solutions
Class 9 - MathematicsChapter 9: CIRCLES
Class 9 - Mathematics
EXERCISE 9.1
Question. 1
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.
Answer:
Prove exactly as Theorem 9.1 by considering chords of congruent circles.
Question. 2
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Answer:
Use SAS axiom of congruence to show the congruence of the two triangles.
EXERCISE 9.2
Question. 1
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.
Answer:
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.
Question. 2
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.
Answer:
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.
Question. 3
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.
Answer:
Proceed as in Example 2.
Question. 4
If a line intersects two concentric circles with centre O at A, B, C and D, prove that AB = CD.
Answer:
Draw perpendicular OM on AD.
Question. 5
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?
Answer:
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.
Question. 6
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.
Answer:
Use the properties of an equilateral triangle and also Pythagoras Theorem.
EXERCISE 9.3
Question. 1
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.
Answer:
45°
Question. 2
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.
Answer:
150°, 30°
Question. 3
In Fig. 9.24, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.
Answer:
10°
Question. 4
In Fig. 9.25, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.
Answer:
80°
Question. 5
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.
Answer:
110°
Question. 6
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.
Answer:
∠BCD = 80° and ∠ECD = 50°
Question. 7
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
Answer:
Draw perpendiculars AM and BN on CD (AB ∥ CD and AB < CD). Show ΔAMD ≅ ΔBNC. This gives ∠C = ∠D and, therefore, ∠A + ∠C = 180°.