Class 9 Maths Ch-9 Circles

NCERT Solutions of Class 9 Maths Ch-9 Circles

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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.

Solution

Given: Two congruent circles with centres O and O’ and radii r, which have chords AB and CD respectively such that AB = CD.
To Prove: ∠AOB = ∠CO’D
Proof:
In ∆AOB and ∆CO’D,
AB = CD [Given]
OA = O’C [Each equal to r]
OB = O’D [Each equal to r]
∴ ∆AOB ≅ ∆CO’D [By SSS congruence criteria]
⇒ ∠AOB = ∠CO’D [C.P.C.T.

Question 2:
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Solution

Given: Two congruent circles with centres O & O’ and radii r which have chords AB and CD respectively such that ∠AOB = ∠CO’D.
To Prove: AB = CD
Proof:
In ∆AOB and ∆CO’D,
OA = O’C [Each equal to r]
OB = O’D [Each equal to r]
∠AOB = ∠CO’D [Given]
∴ ∆AOB ≅ ∆CO’D [By SAS congruence criteria]
Hence, AB = CD [C.P.C.T.]

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NCERT Solutions of Class 9 Maths Ch-9 Circles

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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.

Solution

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.

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.

Question 4:
If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 9.12).

Question 5:
Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. 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 6m each, what is the distance between Reshma and Mandip?

Question 6:
A circular park of radius 20m 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.

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NCERT Solutions of Class 9 Maths Ch-9 Circles

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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.

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

Question 3:
In Fig. 9.24, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

Question 4:
In Fig. 9.25, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

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.

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.

Question 7:
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Question 8:
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Question 9:
Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 9.27). Prove that ∠ACP = ∠ QCD.

Question 10:
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.

Question 11:
ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.

Question 12:
Prove that a cyclic parallelogram is a rectangle.

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NCERT Solutions of Class 9 Maths Ch-9 Circles

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Important Questions for Class 9 Maths Ch-9 Circle

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NCERT Solutions of Class 9 Maths Ch-9 Circles

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