Basic Properties of a Circle

Q.

Two circles touch each other externally at C and AB is a common tangent to the circles. Then, ∠ACB=

1. 30⁰

2. 90^o

3. 60⁰

4. 45⁰

Q. What is the angle between the point of contact of a tangent and the radius of a circle?

1. 90 degrees
2. 45 degrees
3. 80 degrees
4. 60 degree

Q. In the given figure, PQRS is a square lawn with side PQ=42 m. Two circular flower beds are there on the sides PS and QR with center at O, the intersection of its diagonals. Find the total area of the two flower beds(shaded parts).

Q.

In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length

Q. Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

Q.

State and prove the tangent segment theorem

Q.

In the given figure, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find :

(i) AB. (ii) the length of tangent PT.

Q.

Chords AB & CD of a circle are parallel to each other and lie on opposite sides of the centre of the circle. If AB = 36 cm, CD = 48 cm and the distance between the chords is 42 cm. Find the radius of the circle.

Q.

AB is a line segment & M is its mid- point. Semi- circles are drawn with AM, MB & AB as diameters on the same side of the line AB. A circle C (o, r) is drawn so that it touches all the three semicircles. Prove that r = 1/6 AB

Q.

In the figure, given below, O is the centre of the circumcircle of triangle XYZ. Tangents at X and Y intersect at point T. Given $\angle XTY={80}^{\circ }$ and $\angle XOZ={140}^{\circ }$, calculate the value of $\angle ZXY$.

1. ${40}^0$

2. ${60}^0$

3. ${90}^0$

4. ${30}^0$

Q.

ABCD is a cyclic quadrilateral PQ is a tangent at B . If $\angle$DBQ = ${65}^{\circ }$ , then $\angle$BCD is

1. 35°

2. 85°

3. 115°

4. 90°

Q.

In the given figure, AB is the diameter of the circle, with centre O, and AT is the tangent calculate the numerical value of x.

1. ${58}^{\circ }$

2. ${98}^{\circ }$

3. ${50}^{\circ }$

4. ${28}^{\circ }$

Q.

In the given figure PQ is a diameter of a circle with centre O and PT is a tangent at P. QT meets the circle at R. If ∠POR = ${72}^{\circ }$, find $\angle$PTR.

Q.

A tangent PQ at a point P of the circle with radius 5cm meets a line through the centre O at a point Q so that OQ is 12cm. the length of PQ is

1. 8.5 cm

2. √119 cm

3. 12 cm

4. 13 cm

Q. Question 9
If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the point P and Q, prove that PQ is a diameter of the circle.

Q.

What is the distance between two parallel tangents of a circle of the radius 4 cm?

1. 4

2. 0

3. 2

4. 8

Q.

In the given figure, AB is the diameter of the circle. Find the value of $\angle$ ACD

Q.

In the adjoining figure, O is the centre of the circle and AB is a tangent to it at point B. $\angle BDC={65}^o$. Find $\angle$ BAO.

Q.

AB & CD are two equal chord of circle with centre o. If AB and CD, on being produced, meet at a point P outside the circle, Prove that: (i) PA = PC (ii) PB = PD

Q.

In the adjoining figure O is the center of the circle. $\angle$AOD = ${120}^{\circ }$. If the radius of the circle be 'r', then find the sum of the areas of quadrilaterals AODP and OBQC:

1. √3/2 r²

2. 3√3 r²

3. √3 r²

4. None of these

Q.

Find the value of $\angle$ DCE:

1. 100°

2. 80°

3. 90°

4. 75°

Q.

In the figure AB & CD are 2 || chords & O is the centre. If the radius of the circle is 15 cm, find the distance MN between the two chords of length 24cm & 18cm respectively.

1. 20 cm

2. 15 cm

3. 9 cm

4. 25 cm

5. 21 cm

Q.

In a triangle ABC, the incircle (centre O) touches BC. CA and AB at points P, Q and R respectively, Calculate :

$\angle QOR=2\angle QPR$
given that $\angle A={60}^o$

Q.

In the given figure O is the center of the circle and $\angle$BAC = ${25}^{\circ }$ , then the value of $\angle$ADB is :

1. 40°

2. 55°

3. 50°

4. 65°

Q.

AB is diameter and AC is a chord of a circle such that $\angle BAC={30}^{\circ }$. If the tangent at C intersects AB produced in D, prove that BC = BD [4 MARKS]

Q.

In fig., PA is a tangent to a circle of radius 6 cm and PA = 8 cm, then length of PB is

1. 16 cm

2. 18 cm

3. 12 cm

4. 10 cm

Q.

In the given figure, PT touches the circle with centre O at point R. Diameter SQ is produced to meet the tangent TR at P.

Given $\angle SPR=x^o$ and $\angle QRP=y^o$

prove that:

(i) $\angle ORS=y^o$

(ii) write an expression connecting x and y.

Q. In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then the radius of the circle is [CBSE 2011, 12]
(a) 10 cm
(b) 12 cm
(c) 13 cm
(d) 15 cm

Q.

If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at an angle of 80^o then $\angle POA$ is equal to

1. 70^o

2. 60^o

3. 80^o

4. 50^o

Q.

In the figure chords AB & CD when extended meet at X. Given AB = 4 cm, BX = 6 cm, XD = 5 cm calculate the length of CD.

1. 6 cm

2. 7 cm

3. 5 cm

4. 10 cm

5. 14 cm