## Ncert solutions for Maths  class 9 --Circles

A circle is collection of all the points in a plane which are equidistant from a fixed point in the plane

Equal chords of a circle subtend equal angles at the centre

The angle subtended by the two chords of a circle at the centre are equal then the chords are equal

The perpendicular from the centre of a circle to a chord bisects the chord

Equal chords of a circle are equidistant from the centre and vice versa

Angles in the same segment of a circle are equal

Angle subtended by an Arc at the centre is double the angle subtended by it at any point on the remaining part of the circle

Angle in a semicircle is a right angle

The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree

Sum of pair of opposite angles of a quadrilateral is 180 degree then the quadrilateral is cyclic

Ex 10.1

Que 1

Fill in blanks

1) The centre of the circle lies in the------ of the circle

Ans-- interior

2) A point whose distance from the centre of a circle is greater than its radius lies in the ------ of the circle

Ans Exterior

3) The longest chord of a circle is   ------ of the circle

Ans Diameter

4) An Arc is a----- when its ends are the ends of a diameter

Ans semicircle

5) Segment of a circle is the region between an Arc and ------ of the circle

Ans  chord

6) A circle divides the plane on which it lies in ----- parts

Ans --- three

Que 2 True and False

1) Line segment joining the centre to any point on the circle is a radius of the circle. True

2) A circle has only finite number of equal chords. False

3) If a circle is divided into three equal Arc each is a major arc. False

4) A chord of a circle which is twice as long as its radius is a diameter of a circle. True

5) Sector is the region between the chord and its corresponding arc False

6) A circle is a plane figure  True

Exercise  10.2

Que 1

Recall the two circles are congruent if they have the same radii prove that equal chords of a congruent circles subtend equal angles at their centres

Solution---

Que 2

Prove that if chords of a congruent circles substance equal angles at their centres then the chords are equal

Solution--

Exercise 10 .4

Question

How to find the length of the common chord
There are two circles of radius 3 cm and 5 cm they intersect each other the distance between their centre is 4 cm find the length of the common chord

Solution

Que 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 other chord

Que 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 chord

Que 4

Line intersects two concentric circles with centre O at A,B,C and D  prove that AB=CD

Que 5

Three girls Reshma Salma and Mandeep are playing a game by standing on a circle of radius 5 metre drawn in a park throws a ball to the Salma Salma to Mandeep Mandeep to Reshma if the distance between Reshma and Salman and between Salman Mandeep is 6 m each  what is the distance between Reshma in Mandip?

Que 6

Circular park of radius 20m is situated in a colony three boys Ankur Syed and David all sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other find the length of a string of each phone?

Exercise 10.5

Que 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

Solution

Que 3

∆ PQR =100° where P,Q,R are points on a circle with centre O Find angle OPR

Solution

Que 1

A,B,C are three points on a circle with centre O such that angle BOC =30° and angle AOB = 60° If D is a point on the circle other than the arc ABC find angle ADC

Solution

Que 4

Angle ABC =69° ,angle ACB = 31° find angle BDC

Solution

Que 5

A,B,C and D are four points on a circle AC and BD intersect at a point E such that angle BEC =130° and angle ECD = 20° Find angle BAC

Solution

Que 6

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E If angle DBC =70° angle BAC is 30 ° find angle BCD further if AB = BC find angle ECD

Solution

Que 7 if diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral prove that it is a rectangle

Que 8

If the non parallel sides of a trapezium are equal prove that it is cyclic

Que 9

Two circles intersect at two points p and c through b two line segments are drawn to intersect the circles respectively prove that

£ACP=£QCD

Que 10

Circles are drawn taking two sides of a triangle is a diameter prove that the point of intersection of these circles lie on the third side

Que 11

ABC and ADC are the two right triangles with common hypotenuse AC  prove that

Que 12

Prove that a cyclic parallelogram is a rectangle

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