Radius and Tangents of a Circle Are Perpendicular

Q. If PA and PB are tangents from an outside point P, such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

1. 9 cm
2. 10 cm
3. 11 cm
4. 8 cm

Q. In figure, PQ is a chord of a circle with centre O and PT is a tangent. If $\angle QPT={60}^{\circ },$, find $\angle PRQ$.

Q.

If two chords having lengths ${\mathrm{a}}^2-1$ and $3\left(\mathrm{a}+1\right)$, where $\mathrm{a}$ is a constant of a circle bisect each other, then the radius of the circle is

1. $6$

2. $\frac{15}{2}$

3. $8$

4. $\frac{19}{2}$

5. $10$

Q. Question 7
In figure, if 0 is the centre of a circle, PQ is a chord and the tangent PR makes an angle of ${50}^{\circ }$ with PQ, then $\angle$ POQ is equal to
(A) 100°
(B) 80°
(C) 90°
(D) 75°

Q. Question 3
Prove that the centre of circle touching two intersecting lines lies on the angle bisector of the lines.

Q. From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA = 14 cm, find the perimeter of ∆PCD.

1. 14 cm
2. 18 cm
3. 24 cm
4. 28 cm

Q.

If TP and TQ are the two tangents to a circle with center O such that ∠POQ = 115^o, then ∠PTQ is equal to

1. 65^o

2. 15^o

3. 90^o

4. 55^o

Q.

A tangent is drawn at a point P on a circle. A line through the centre O of the circle of radius 7 cm cuts the tangent at Q such that PQ = 24 cm. Find OQ.

1. 10 cm

2. 15 cm

3. 20 cm

4. 25 cm

Q.

The angle between two tangents to a circle may be $0°$. Write ‘True’ or ‘False’ and justify your answer.

1. True
2. False

Q. Find the length of the tangent to a circle with centre O and radius = 5 cm, from a point Q such that OQ= 13 cm.

1. 13 cm
2. 5 cm
3. 9 cm
4. 12 cm

Q.

In the given figure, PAQ is the tangent. BC is the diameter of the circle. If $\angle BAQ={60}^{\circ }$, then $\angle ABC$ = _____.

1. ${25}^{\circ }$

2. ${30}^{\circ }$

3. ${45}^{\circ }$

4. ${60}^{\circ }$

Q. In the given figure PQ and PR are two tangents on a circle with centre O. If $\angle QPR={60}^{\circ }$, then $\angle QRO={30}^{\circ }.$

1. True
2. False

Q. Two concentric circles of radii 5 cm and 3cm are drawn. Find the length of the chord of the larger circle which touches the smaller circle.

1. 10 cm
2. 8 cm
3. 4 cm
4. 12 cm

Q.

Two tangents are drawn from a point P to a circle touching the circle at A and B respectively . If PA = 6 cm and AB = 3 cm, then find the perimeter of triangle ABP .

1. 36 cm

2. 12 cm

3. 30 cm

4. 15 cm

Q. tangent/tangents can be drawn from a point on the circle.

1. Atmost two
2. Only one
3. Atleast two
4. Infinite

Q. Two parallel chords are drawn on either side of the centre of a circle of diameter 30 cm. If the length of one chord is 24 cm and the distance between the two chords is 21 cm, then find the length of another chord.

Q.

In the given figure, PA and PB are two tangents to the circle with centre O. If $\angle APB=50°,$ then what is the measure of $\angle OAB?$

1. 25°
2. 30°
3. 35°
4. 40°

Q. tangent/tangents can be drawn from a point on the circle.

1. Atmost two
2. Only one
3. Atleast two
4. Infinite

Q.

The number of tangents that can be drawn from a point inside the circle is/are ____.

1. zero
2. two
3. four
4. one

Q. In the given figure, if PQ is the diameter, then x is equal to ?

1. ${30}^0$
2. ${45}^0$
3. ${70}^0$
4. ${60}^0$

Q. In the given figure, AB is a diameter and AC is a chord of the circle and the tangent at C intersects AB produced in D. If $\angle BAC={30}^{\circ }$, then find the value of $\angle CDB$

Q.

Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Then, find the relation between $\angle CPA$ and $\angle DPB$ .

Q. If the angle of the major arc increases then the angle of the minor arc _____.

1. Decreases
2. Remains same
3. Can't say
4. Increases

Q.

O is the centre of the circle as shown in the figure, $\angle ORP={35}^{\circ }$ and the distance between P and Q through 'O' is 4 cm. What is the measure of $\angle ROQ$?

1. ${35}^{\circ }$

2. ${55}^{\circ }$

3. ${105}^{\circ }$

4. ${70}^{\circ }$

Q. A restaurant POQR is being constructed in the middle of a circular lake. The circle, which has its center at O is touching the mid-points of the boundary ABCD as shown in the figure. If AB=CD and AD=BC, then

Quadrilateral ABCD can be

1. square
2. rectangle
3. rhombus
4. parallelogram

Q.

The tangent is ____ to the radius at the point of contact.

1. parallel

2. perpendicular

3. collinear

4. None of the above

Q. In the given figure AB = BC, $\angle ABC={68}^o$ DA and DB are the tangents to the circle with centre O.
Calculate the measure of $\angle ADB$

Q.

Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Then, find the relation between $\angle CPA$ and $\angle DPB$ .

Q. If PQ and PR are tangents to a circle with centre 'O' from an external point P, then sum of $\angle$ PQO and $\angle$ PRO is____.

1. ${90}^{\circ }$
2. ${180}^{\circ }$
3. ${270}^{\circ }$
4. can't be determined.

Q.

In the given diagram, a circle is inscribed in a right angled triangle such that AF = 6 cm and EC = 15 cm. Find the difference between CD and BD.

1. 3 cm

2. 1 cm

3. 2 cm

4. 4 cm