Tangent from an External Point Are Equal

Q.

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

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

Q. In fig. 1, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and $\angle CAB={30}^{\circ }$, find $\angle PCA.$

Q.

PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∠POR=120⁰, then ∠OPQ is

1. 90⁰

2. 45⁰

3. 30⁰

4. 60⁰

Q. If PT is tangent drawn from a point P to a circle touching it at T and O is the centre of the circle, then ∠OPT + ∠POT =

(a) 30°

(b) 60°

(c) 90°

(d) 180°

Q. Two tangents TP and TQ are drawn from an external point T to a circle with centre O as shown in Fig. 10.73. If they are inclined to each other at an angle of 100°, then what is the value of ∠POQ?

Q. State True or False
The lengths of tangents drawn from an external point are equal.

1. True
2. False

Q.

PT is a tangent from an external point P and PAB is a secant. If PT = 6cm and PA = 3cm, then the length of PB is __

1. 2

2. 12

3. 6

4. 36

Q.
In the figure given above, PA and PB are tangents to the circle from point P. How many real values of x exist such that the length of PA is x and length of PB is $x^2+1$?

1. 0
2. 1
3. 2
4. 3

Q. In Fig. 1, PQ and PR are tangents to the circle with centre O such that ∠QPR = 50°,

(a) 25°

(b) 30°

(c) 40°

(d) 50°

Q.

TA and TB are tangents to a circle with centre O from an external point T. OT intersects the circle at point P. Then, AP bisects the $\angle TAB$.

1. False

2. True

Q. PQ is a tangent to a circle with centre O at the point P. If Δ OPQ is an isosceles triangle, then ∠OQP is equal to

(a) 30°

(b) 45°

(c) 60°

(d) 90°

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-segment joining the points of contact at the centre.

Q. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that sums of opposite sides are equal.

Q.

A circle may have…….

1. 1 tangent

2. 2 tangent

3. Infinite tangents

4. No tangent

Q.

In the given figure, a circle touches the side BC of $\Delta$ABC at P and AB and AC produced at Q and R respectively. If AQ = 15 cm, find the perimeter of $\Delta$ABC.

__

Q.

The lengths of tangents drawn from an external point to a circle are equal.

1. False

2. True

Q.

Maximum number of tangents to a circle which are parallel to a secant can be two.

1. 2

2. 1

Q.

In the given figure, AB is a tangent to the circle with centre O. If OP = PC, then find $\angle OCP$.

[1 Mark]

Q.

The incircle of triangle ABC touches sides AB, BC and CA at P, Q and R respectively. If AP = BQ = CR = 2 cm, then which of the following statements are true?

1. Perimeter of $\Delta ABC$ = 12 cm

2. $\angle B={60}^{\circ }$

3. BP + CQ + AR = 6 cm

4. $△ABC$ is equilateral

Q.

The lengths of tangents drawn from a point outside a circle are not equal.

1. True

2. False

Q. If two tangents drawn from an external point P touches the circle at points Q and R, then the triangle PQR is a/an triangle.

1. Equilateral
2. Isosceles
3. Scalene
4. Right-angled

Q.

Find the length of the tangent AP

1. 8 cm

2. 12 cm

3. 6 cm

4. 10 cm

Q. The lengths of tangents drawn from an external point to a circle are

1. Equal
2. Unequal
3. Depends upon the radius of the circle
4. None of these

Q.

In the adjoining figure, if AD, AE and BC are tangents to the circle at D, E and F respectively. Then,

1. $AD=AB+BC+CA$

2. $2AD=AB+BC+CA$

3. $3AD=AB+BC+CA$

4. $4AD=AB+BC+CA$

Q. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

Q.

In the figure, PQ is a tangent at a point R of the circle with centre O. If ∠TRQ = 30^o.The measure of ∠PRS is

1. 120^o

2. 90^o

3. 45^o

4. 60^o

Q. Assertion (A)
If two tangents are drawn to a circle from an external point, they subtend equal angles at the centre.

Reason (R)
A parallelogram circumscribing a circle is a rhombus.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Q.

The lengths of tangents drawn from an external point to a circle are equal.

1. False

2. True

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 $\angle ZXY$.

Q. Okay, now this guy is stuck. You need to help him out. What is the first thing to identify?

1. Lengths of tangents drawn from an external point are equal.
2. Radius is perpendicular to the tangent at the point of contact.
3. The length of radius is same at any point on the circumference of the circle.
4. The line joining the external point and the centre of the circle bisects the angle between the tangents.