Radius and Tangent to a Circle Are Perpendicular

Q.

Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle, is bisected at the point of contact.

Q. Question 10
In figure. If PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and $\angle BQR={70}^{\circ }$ , then $\angle$AQB is equal to
(A) 20°
(B) 40°
(C) 35°
(D) 45°

Q. Question 14
A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q . If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ∆ABC.

Q. A piece of wire 22 cm long is bent into the form of an arc of a circle subtending an angle of 60° at its centre. Find the radius of the circle. ​

Q. Question 9
In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that $\angle AOB={90}^{\circ }$.

Q. Given the length of tangent and the distance of external point from the centre, match the corresponding radii.

1. 9 cm
2. 7 cm
3. 8 cm
4. 5 cm

Q. From a point P outside the circle, with centre O, tangents PA and PB are drawn.Then ∠AOP is equal to

1. ∠APO
2. ∠PAO
3. ∠BOP
4. ∠APB

Q.

Two tangents PA and PB are drawn to the circle with centre O, such that $\angle$APB $={120}^{\circ }$. What is the relation between OP and AP?

1. $OP=\frac{1}{2}AP$

2. $AP=\frac{2}{3}OP$

3. $OP=2AP$

4. $OP=AP$

Q. In a shop, there were 4 dolls of different heights A, B, C and D. D is neither as tall as A nor as short as C. B is shorter than D but taller than C. If Mani wants to purchase the tallest doll, which one should she purchase?

1. Either A or D
2. Either B or D
3. Only D
4. Only A

Q. Question 10
In figure. If PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and $\angle BQR={70}^{\circ }$ , then $\angle$AQB is equal to
(A) 20°
(B) 40°
(C) 35°
(D) 45°

Q. Given the length of tangent and the distance of external point from the centre, match the corresponding radii.

1. 9 cm
2. 7 cm
3. 8 cm
4. 5 cm

Q.

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Q. Question 5
Two circles with centers O and O’ of radii 3 cm and 4 cm , respectively intersect at two points P and Q, such that OP and O’P are tangents to the two circles. Find the length of the common chord PQ.

Q. Perpendicular from the centre of a circle to a chord divides the chord in the ratio 2:1.

1. False
2. True

Q. E is a point outside the circle as shown. ME and NE are tangents drawn to the circle from this point. Quadrilateral OMEN is symmetrical with respect to OE. State whether true or false.

1. True
2. False

Q. Question 12
The tangent at a point C of a circle and a diameter AB when extended intersect at P. If $\angle PCA={110}^{\circ }$ , find $\angle$ CBA.

Q. It is possible to draw tangents to a circle from an external point, even when the centre of the circle is not known.

1. True
2. False

Q.

Two tangents PA and PB are drawn to the circle with centre O, such that $\angle$APB $={120}^{\circ }$. Prove that
OP = 2AP ?

Q. The relation between length of sides of triangle and segment's length from vertex to a point on opposite side. It is referred as:

1. Pythagoras theorem
2. Apollonius theorem
3. None of the above
4. Law of tangents

Q. In the given figure RTP and STQ are common tangents to the two circles with centres A and B. The radii of the two circles are 3 cm long are 5 cm long respectively.
If ST : TQ =1 : 3 and RT = 4 cm, calculate (a) the length of-QT (b) the length of AB

Q. Given a point outside a circle, construct tangents through the point to the circle.

Q.

In the given figure, a circle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14 cm, BC = 8 cm and CA = 12 cm. Find the lengths AD, BE and CF.

Q.

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of ${60}^0$. Find the value of $\angle$ AOB.

1. ${120}^{\circ }$

2. ${80}^{\circ }$

3. ${40}^{\circ }$

4. ${30}^{\circ }$

Q. Question 5
Two circles with centers O and O’ of radii 3 cm and 4 cm , respectively intersect at two points P and Q, such that OP and O’P are tangents to the two circles. Find the length of the common chord PQ.

Q. Perpendicular bisector of any chord of a circle passes through the centre of the circle.

1. False
2. True