Tangents from External Points

Q. Question 3
In figure, AB is a chord of the circle and AOC is its diameter such that $\angle ACB={50}^{\circ }$. If AT is the tangent to the circle at the point A, then $\angle$ BAT is equal to
(A) 65°
(B) 60°
(C) 50°
(D) 40°

Q.

The length of the tangent drawn from a point $8cm$ away from the centre of a circle of radius $6cm$ is $2\sqrt{7}cm.$

1. True

2. False

Q.

In the figure, O is the center of the circle, PQ is tangent to the circle at A. If $\angle PAB={58}^{\circ }$, find $\angle ABQ$ and $\angle AQB$. [2 MARKS]

Q.

If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

Q.

In the figure, AQ, AR and BC are tangents to circle with centre O, if AB = 7 cm, BC = 5 cm, AC = 5 cm, then length of the tangent AQ is:

1. 17 cm

2. 5 cm

3. 7 cm

4. 8.5 cm

Q.

In the given figure, PT touches a circle whose centre is O at R. Diameter SQ produced meets PT at P. If $\angle SPR=x^{\circ }$ and $\angle QRP=y^{circ}$, then which of the following statements relate x and y?

1. $x+2y={90}^{\circ }$
   

2. $x+y={90}^{\circ }$
   

3. $x+2y={45}^{\circ }$
   

4. x=2y

Q.

Two circles of radii 8 cm and 5 cm with their centres A and B respectively touch externally as shown in the figure. Calculate the length of direct common tangent PQ

Q.

If tangents PA and PB from a point P to a circle with centre O are inclined each other an angle of 70^o, then find ∠POA

1. 65^o

2. 50^o

3. 55^o

4. 60^o

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. 12 cm

2. 15 cm

3. 30 cm

4. 36 cm

Q. The length of three concesutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm, and 7 cm respectively. Determine the length of the fourth side.

Q.

A tangent PA is drawn from an external point P to a circle of radius 3√2 cm such that the distance of the point P from O is 6 cm as shown figure. The value of ∠APO is

1. 60^o

2. 45^o

3. 75^o

4. 30^o

Q. Question 11
Prove that the parallelogram circumscribing a circle is a rhombus.

Q.

Two parallel tangents of a circle meet a third tangent at points P and Q. Then, PQ subtends a right angle at the centre.

1. False

2. True

Q.

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

Q.

In the figure, the pair of tangents AP and AQ drawn form an external point A to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. Then the radius of the circle is:

1. 10 cm

2. 7.5 cm

3. 5 cm

4. 2.5 cm

Q.

From a point P outside a circle, with centre O, tangents PA and PB are drawn. Then, select the statements that are true.

1. $\angle AOP=\angle BOP$

2. OP is the $\perp$ bisector of chord AB.

3. $\angle AOP=2\angle BOP$

4. $\angle AOP=7\angle BOP$

Q.

The length of tangent drawn from an external point P to a circle, with centre O, is 8 cm. If the radius of the circle is 6 cm, then the length of OP (in cm) is:

1. 2√7 cm

2. 4√7 cm

3. 10.5 cm

4. 10 cm

Q. In Q.No. 1, if PB = 10 cm, what is the perimeter of Δ PCD?

Q.

In the figure, if from an external point T, TP and TQ are two tangents to a circle with centre O so that ∠POQ = 110^o, then, ∠PTQ is:

1. 60^o

2. 90^o

3. 70^o

4. 80^o

Q. Find the length of tangent drawn to circle with radius 8 cm from a point 17 cm way from the centre of the circle

Q.

The tangent to a circle is perpendicular to the radius at the ___.

1. Point of secant

2. Point of contact

3. End Point of secant

4. Tangent

Q.

Quadrilateral ABCD circumscribes a circle as shown in figure. The side of the quadrilateral which is equal to AP + BR is:

1. AC

2. AD

3. AB

4. BC

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.

Q. Prove that the tangent of any point of a circle is perpendicular to the radius through the point of contact.

Q.

If the sides of a quadrilateral ABCD touch a circle, then AB+CD=BC+AD

1. True

2. False

Q.

Two circles of radii 8 cm and 5 cm with their centres A and B respectively touch externally as shown in the figure. Calculate the length of direct common tangent PQ

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 = 2 cm then, find the perimeter (in cm) of quadrilateral PQOR.

1. 14

Q.

All the sides of the quadrilateral ABCD are tangential to the circle with centre M. What is the perimeter of the quadrilateral ABCD?

1. 20 cm

2. 80 cm

3. 15 cm

4. 34 cm

Q.

Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. Then, which of the following statements are true?

1. $\angle PAQ=9\angle OPQ$
2. $\angle PAQ=2\angle OPQ$
3. $\angle OPQ=2\angle PAQ$
4. $\angle PAQ=4\angle OPQ$

Q.

All the sides of the triangle ABC are tangential to the circle with centre R. What is the perimeter of $\Delta ABC$?

1. 28 cm

2. 20 cm

3. 5 cm

4. 30 cm