Tangents Drawn from an External Point

Q.

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the tangents BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.

Q. The maximum number of tangents that can be drawn to a circle from an external point is ___.

Q. What the distance between two parallel tangents to a circle of radius 5 cm?

Q.

A circle is inscribed in a triangle with sides $3,4$ and $5cm.$ The radius of the circle is ___ $cm.$

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

(a) 30º (b) 45º (c) 60º (d) 90º [CBSE 2014]

Q. Consider the following statements and choose the correct option.
1) A tangent is a line that touches the circle at one point.
2) A tangent is a special case of the secant where the points of intersection coincide.

1. 1 is false and 2 is true
2. Both 1 and 2 are true
3. Both 1 and 2 are false
4. 1 is true and 2 is false

Q. Two tangents PA and PB are drawn from an external point P to a circle with centre O. Prove that AOBP is a cyclic quadrilateral.

Q.

In the given figure, TAS is a tangent to the circle, with centre O, at the point A. If $\angle$OBA = ${32}^{\circ }$, find the value of x.

Q. What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.

Q.

In the figure given below, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively.

Given that AB = 8 cm, calculate PQ.

1. 16 cm

2. 15 cm

3. 19 cm

4. 25 cm

Q. In Fig. 3, two tangents PQ are PR are drawn to a circle with centre O from an external point P. Prove that ∠QPR = 2 ∠OQR.

Q. Two tangents making an angle of 120° with each other are drawn to a circle of radius 6 cm, then the length of each tangent is equal to

(a) $\sqrt{3}cm$

(b) 6 $\sqrt{3}cm$

(c) $\sqrt{2}cm$

(d) 2 $\sqrt{3}cm$

Q. If PT is a tangent at T to a circle whose centre is O and OP = 17 cm, OT = 8 cm, Find the length of the tangent segment PT.

Q.

In the given fig., angle OBC = 30^o, then value of x is:

1. 15^o

2. 110^o

3. 100^o

4. 30^o

Q.

In the given figure, AB is the diameter of a circle and P is a point on AB extended.

A tangent from P touches the circle at Q. If PQ = 4 cm and PA = 8 cm ,

then the radius of the circle is ____ cm.

1. 3.5 cm

2. 4 cm

3. 3 cm

4. 2 cm

Q.

In the following figure, a circle is inscribed in the quadrilateral ABCD. If BC = 38 cm. QB = 27 cm, DC = 25 cm and that AD is perpendicular to DC, find the radius of the circle.

1. 34 cm

2. 24 cm

3. 13 cm

4. 14 cm

Q.

In the figure, PM is a tangent to the circle and PA = AM. Find PA.PB = ?

1. MB²

2. 1.5 MB

3. 2 MB

4. MB

5. 1.75 MB

Q. What is the value of angle QPR in degrees?

1. 30
2. 60
3. 120
4. 20

Q. In an isosceles triangle ABC, AB = BC, ∠B = 20◦. M, N are on AB and BC respectively such that ∠MCA = 60◦, ∠NAC = 50◦. Find ∠NMC in degrees.

Q. Theorem: Tangent segments drawn from an external point to a circle are congruent

Q. In the given figure, seg EF is a diameter and seg DF is a tangent segment. The radius of the circle is r. Prove that, DE × GE = 4r²

Q. In Fig . 10.69, the tangent at a point C of a circle and a diameter AB when extended intersect at P . If $\angle$PCA =110⁰, find $\angle$CBA. [Hint: Join CO.]
figure

Q.

The length of the tangent to the circle from an exterior point P is 5cm. What is the length of another tangent drawn to the same circle from the exterior point P?

Q.

From an external point P, two tangents are drawn that touch the circle at points Q and R. The centre of the circle is O. Points O and P are joined. The ratio of angles OPR and OPQ is

1. 1:1

2. 2:1

3. 3:1

4. 4:1

Q. To draw a pair of tangents to a circle which are inclined to each other at an angle of ${45}^{\circ }$, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be

1. $135$
2. $60$
3. $90$
4. $45$

Q.

Prove that the least perimeter of an isosceles triangle in which a circle of radius $r$ can be inscribed is $6\sqrt{3r}$.

Q. The maximum number of tangents that can be drawn from an external point to a circle is:

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

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 given figure, AP and AQ are the tangents drawn to a circle from a point A outside the circle.
If $\angle$PAQ = ${50}^{\circ }$ then, find $\angle$OPQ.

1. ${100}^{\circ }$
2. ${50}^{\circ }$
3. ${25}^{\circ }$
4. ${12.5}^{\circ }$

Q. In the figure given below, PA and PB are tangents to the circle from point P. Which of the following is the correct relation between PA and PB?

1. $PA>PB$
2. $PA=PB$
3. $PA<PB$
4. $P{A}^2=P{B}^2+r^2$