Summary: Tangents and Its Properties

Q. In the given figure, O is the centre of the circle $C_1$. If CD and CB are the tangents to the circle $C_1$ at point D and B respectively, then $\angle BDC$ is equal to

1. 
   ${70}^{\circ }$
2. 
   ${60}^{\circ }$
3. 
   ${50}^{\circ }$
   
4. 
   ${80}^{\circ }$

Q. In the given figure, FDE is the tangent to the circle at point D. If $\angle DAB={56}^{\circ }and\angle DBC={30}^{\circ }$, then the measure of $\angle CDF+\angle BDC$ is equal to

1. 
   ${124}^{\circ }$
2. 
   ${134}^{\circ }$
3. 
   
   ${176}^{\circ }$
4. 
   ${156}^{\circ }$

Q.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Q. In the given figure, XY is a tangent to the circle with centre O at A. If $\angle CAX=\angle BAY={60}^{\circ }$, then OD equals

1. 
   $\frac{1}{3}OA$
2. 
   $\frac{2}{3}OA$
3. 
   $DE$
4. 
   $\frac{2}{3}DE$

Q.

If$a,bandc$ be in H.P, then

1. $\frac{a-b}{b-c}=\frac{a}{c}$

2. $\frac{b-c}{c-a}=\frac{b}{a}$

3. $\frac{c-a}{a-b}=\frac{c}{b}$

4. $\frac{a-b}{b-c}=\frac{c}{a}$

Q.

In a$∆PQR$, $P$ is the largest angle and$\cos P=\frac{1}{3}$, Further in the circle of the triangle touches the sides$PQ$, $QR$, and $RP$ at $N$, $L$, and $M$ respectively, such that the lengths of $PN$, $QL$, and $RM$ are consecutive even integers, Then, possible length(s) of the side (s) of the triangle is (are)

1. $16$

2. $18$

3. $24$

4. $22$

Q.

The two tangents from an external point P to a circle with centre at O are PA and PB. If ∠APB=70°, then the value of ∠ AOB is:(2 mark)

1. 130°
2. 120°

3. 100°

4. 110°

Q.

Two circles touch each other externally at$P$. $AB$ is a common tangent to the circle touching them at $A$ and $B$. What is the value of angle formed at $APB$$?$

1. $30°$

2. $45°$

3. $60°$

4. $90°$

Q. In a circle, a tangent PT is drawn from a point P which touches at point T to the circle and a secant PAB is drawn which intersects the circle at A and B. If PT = 15 cm and AB = 16 cm, then the length of PA is

1. 
   25 cm
2. 
   16 cm
3. 
   15 cm
4. 
   9 cm

Q. In the given figure if C is the centre of the circle and $\angle PQC={25}^0$ and $\angle PRC={15}^0$, then $\angle QCR$ is equal to

1. ${40}^0$
2. ${60}^0$
3. ${80}^0$
4. ${120}^0$

Q. In the given figure, PQ and RS are two chords of a circle intersecting each other at a point T outside the circle. If PT = 3 cm, PQ = 5 cm, RS = 2 cm, then the length of RT is

1. 
   2 cm
2. 
   3 cm
3. 
   4 cm
4. 
   6 cm

Q. In the given figure, ABC is a triangle, a circle with diameter BC is drawn, intersecting AB and AC at D and E respectively. If the lengths of AB, AC and CD are 30 cm, 25 cm and 20 cm respectively, then the sum of lengths of AT and BE is

(AT is tangent to the circle)

1. 
   $3\sqrt{2}(5+4\sqrt{2})cm$
2. 
   $3\sqrt{2}(4+5\sqrt{2})cm$
3. 
   $3(5+4\sqrt{2})cm$
4. 
   $6(5+4\sqrt{2})cm$

Q. In the given figure, ABCD is a kite $(BC>AB)$ inside the circle given below, AP and CQ are the tangents to the circle at A and C respectively. If $\angle DAB:\angle DCB=11:7$, then the value of $\frac{\angle BCQ+\angle DAP}{{360}^{\circ }}$ is

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

Q. In the given figure, PCD is a secant to the circle with centre O from a point P outside the circle and PA is a tangent. If PA = 12 cm and PC = 7.2 cm, then the radius of the circle is

1. 
   16 cm
2. 
   4cm
3. 
   8cm
4. 
   6 cm

Q. In the given figure, CD and AB are intersecting chords and EF is a tangent to the circle. If OD = 4 cm, OB = 12 cm, OA = 4 cm and DE = 4 cm, then the length of EF is equal to

1. 
   $10\sqrt{5}$
2. 
   $4\sqrt{5}$
3. 
   $2\sqrt{5}$
4. 
   $6\sqrt{5}$

Q. In the given figure, PQR is a triangle in which PQ = PR. A circle is drawn which passes through Q and touches PR at S and intersects PQ at T. If S is the mid-point of PR and PT = 4.5 cm, then length of PQ is

1. 
   9 cm
2. 
   13.5 cm
3. 
   18cm
4. 
   22.5 cm

Q. MN and RS are chord of a circle, which intersect at P outside the circle. If PN = 3cm, MN= 5 cm and PR = 2cm, then the value of SR is equal to

1. 5 cm
2. 8 cm
3. 15 cm
4. 10 cm

Q. In the given figure, O is the centre of the circle, AB and CD are the chords of the circle which intersect at P. If $\text{PA = 4 cm, PB = 6 cm, OB = 7 cm and CD}$ is 1 cm greater than twice of OP, then the difference between the length of $PDandPC$ is

1. 
   3 cm
2. 
   4 cm
3. 
   5 cm
4. 
   6 cm

Q. In a circle of radius 13 𝑐𝑚, two chords MN and JK are at a distance of 5 𝑐𝑚 from the centre. Find the value of
(MN + 2 × JK). [4 Marks]

Q. In a triangle $ABC$, right angled at $B,BC=15cm$ and $AB=8cm$. A circle is inscribed in triangle $ABC$. Then the radius of the circle is

Q. In the given figure, AB = 3 cm, DE = 5 cm. If AD = 4 cm, then BC is equal to

1. 
   $\frac{10}{3}cm$
2. 
   $9cm$
3. 
   $10cm$
4. 
   $\frac{16}{3}cm$
   

Q. Choose the area enclosed between the secant and the tangent.

1. 
2. 
3. 
4. 

Q. Two circle touch externally at a point P. From a point T on a tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR.

Q. If $AP$ and $AQ$ are the two tangents a circle with centre $O$ so that $\angle POQ={110}^o$ , then $\angle PAQ$ is equal to :

1. ${60}^o$
2. ${70}^o$
3. ${80}^o$
4. ${90}^o$

Q. In the shown diagram, if the angle between two chords AB and AC is ${65}^0$, then the angle between two tangents which are drawn at B and C is

1. ${50}^0$
2. ${30}^0$
3. ${60}^0$
4. ${40}^0$

Q. PQ and RS are chords of a circle which intersect at O outside the circle. If PQ = 5 cm, OQ = 9 cm and RS = 15 cm, then the length of OS is

1. 
   27 cm
2. 
   9 cm
3. 6 cm
4. 
   15 cm

Q. Choose the area enclosed between the secant and the tangent.

1. 
2. 
3. 
4. 

Q. Draw a circle with center $O$ and radius $6$cm. Take a point $P$ outside the circle at a distance of $10$ cm from $O$. Draw tangents to the circle from point $P$. Let the tangents intersect the circle in points $A$ and $B$. Find the approximate value of $\angle BOP$ in degrees.

1. None of these
2. ${53}^o$
3. ${37}^o$
4. ${45}^o$

Q. $\text{From the given figure, find}\angle AOB\text{(in degrees )}.$

1. 140

Q.
$\text{If TP and TQ are two tangents to a circle}\text{with center O such that}\angle POQ={110}^o,\text{then}\angle PTQ\text{will be}$.

1. ${70}^{\circ }$
2. ${80}^{\circ }$
3. ${90}^{\circ }$