Angle between Tangent and Radius

Q. In the given figure, a circle with centre O is inscribed in a quadrilateral PQRS in which $\angle S={90}^{\circ }$. If PQ = 9 cm, PS = 13 cm and QU = 3 cm, then the radius of the circle is

1. 
   $5cm$
2. 
   $6cm$
3. 
   $7cm$
4. 
   
   $7\sqrt{2}cm$
   

Q.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the center

Q.

$\text{In the given figure, PT is a tangent of a}\text{circle, with centre O, at point R. If}\text{diameter SQ is produced, it meets with}\text{PT at point P with}\angle SPR=x\text{and}\angle QSR=y,\text{then find the value of}x+2y.$

1. ${45}^{\circ }$

2. ${60}^{\circ }$

3. ${90}^{\circ }$

4. ${150}^{\circ }$

Q. In the given figure, a circle is inscribed in a right-angled triangle such that $AF=6\text{cm}$ and $EC=15\text{cm}$. The perimeter of the triangle is:

1. $68\text{cm}$
2. $70\text{cm}$
3. $72\text{cm}$
4. $74\text{cm}$

Q.

In the given figure, AB and BC are the tangents to the circle from the point B. D is the centre of the circle. BD = 5 cm and
CD = 3 cm. Find the value of AB - BC.

1. 4 cm

2. 9 cm

3. 20 cm

4. 0 cm

Q. Question 10
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q. In the given figure, PQ is a chord of a circle with centre O and PT is a tangent. If $\angle QPT=60°,$ find $\angle PRQ.$

1. 90°
2. 100°
3. 120°
4. 110°

Q.

In the given figure, PQ and PR are two tangents drawn from an external point P to a circle with centre 'O'. If OQ = 3 cm and PS = 2cm then, find the perimeter (in cm) of quadrilateral PQOR.

1. 12
2. 10
3. 14
4. 16

Q. If the angles subtented by a minor arc is 30 degree, then what is the angle subtended by the major arc?

1. ${360}^{\circ }$
2. ${330}^{\circ }$
3. ${300}^{\circ }$
4. ${270}^{\circ }$

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 adjoining figure, $M$ is the centre of the circle and seg $KL$ is a tangent segment. If $MK=12$, $KL=6\sqrt{3}$, then find
i. Radius of the circle.
ii. Measures of $\angle K$ and $\angle M$.

Q.

In the given fig. PQ is tangent at point R of the circle with centre O. if $\angle$TRQ = ${30}^0$, find $\angle$PRS

Q. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Q.

A point $P$ is at a distance of $25cm$ from the centre of a circle. The radius of the circle is $7cm$ and length of the tangent drawn from $P$ to the circle is $xcm.$ The value of $x$ is ___ $cm.$

Q. PQ and PR are two tangents drawn from a point P. If centre 'O' of the circle and P are joined, then
$\angle OPR:\angle OPQ$ =.

1. $2:1$
2. $1:1$
3. $1:2$
4. $4:1$

Q.

In the above figure, BC is a tangent at B. A is the centre of the circle. If AB = BC, then find the value of $\angle BAC$

1. ${10}^{\circ }$

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

3. ${20}^{\circ }$

4. ${45}^{\circ }$

Q. $\text{In the given figure, ABCD is a}\text{quadrilateral circumscribing a circle with}\text{centre 'O'. If}\angle B={90}^{\circ },AD=23cm,AB=29cm\text{and}DS=5cm,\text{then find}\text{the radius of the circle.}$

1. 11 cm
2. 12 cm
3. 14 cm
4. 9 cm

Q. Question 6
If angle between two tangents drawn from a point P to a circle of radius a and centre O is ${60}^{\circ }$ , then OP = a $\sqrt{3}$.

Q.

Tangents from a point P are drawn onto a circle with angle between them as ${120}^{\circ }$ . The tangents from P meet the circle at A and B. If a line is drawn from point P through the center of the circle O, then find the measure of $\angle$POA.

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

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

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

4. ${90}^{\circ }$

Q. A point $P$ is at a distance of $29$ cm from the centre of a circle of radius $20$ cm. Find the length of the tangent drawn from $P$ to the circle.

Q.

The radius of a circle is 8 cm. Calculate the length of a tangent drawn to this circle from a point at a distance of 10 cm from its centre.

1. 9 cm

2. 7 cm

3. 2 cm

4. 6 cm

Q. In the given figure, O is the centre of a circle and two tangents KR, KS are drawn on the circle from a point K lying outside the circle. Prove that KR = KS.

Q. From a point $P$ , the length of the tangent to a circle is $15cm$ and the distance of $P$ from the center of the circle is $17cm$ . Then what is the radius of the circle ?

Q.

The centre of a circle is O. There is a line XY which shares one common point with the circle. A line segment OP is drawn from O to line XY such that P is a point on line XY. What happens if P lies on the circle?

1. OP is not perpendicular to XY.

2. OP need not be radius of circle.

3. OP is perpendicular to XY.

4. The angle between OP and XY is variable.

Q. Write True or False and justify your answer in each of the following :

1. Ambiguous
2. Data insufficient
3. False
4. True

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. rhombus
2. square
3. rectangle
4. parallelogram

Q. If two tangents PA and PB are drawn to a circle with centre O from an external point P (figure), then match the column.

Q. If two tangents inclined at an angle of ${60}^{\circ }$, are drawn to a circle of radius 3 cm, then length of each tangent (in cm) is equal to :

1. 3
2. 6
3. $\frac{3}{2}\sqrt{3}$
4. $3\sqrt{3}$

Q. In the given figure, PA and PB are two tangents drawn from an external point P. If $\angle APB={40}^{\circ }$, then $\angle PAB=$

Q. In the given figure, PA and PB are two tangents drawn from an external point P. If $\angle APB={40}^{\circ }$, then $\angle AOB=$

1. ${110}^{\circ }$
2. ${140}^{\circ }$
3. ${290}^{\circ }$