Tangent and Radius

Q.

A circle is inscribed inside a right-angled triangle. If BC = a, CA= b and AB = c, then the radius of the circle is ____.

1. $\frac{(a+b+c)}{2}$

2. a+b+c

3. $\frac{(a+b-c)}{2}$

4. Can't be determined.

Q.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Q.

In the adjoining figure ${}^{\prime }{O}^{\prime }$ is the center of circle, $\angle$CAO $=$ ${25}^{\circ }$ and $\angle$CBO $=$ ${35}^{\circ }$. What is the value of $\angle$AOB?

1. Data insufficient

2. 55°

3. 110°

4. 120°

Q.

In the given figure, PQ is a transverse common tangent to two circles with centers A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.

Q.

Two chords AB and AC of a circle are equal. Then, the centre of the circle, lies on the bisector of angle BAC.

1. False

2. True

Q.

In the given figure, C and D are points on the semi-circle described on AB as diameter. Given angle BAD = ${70}^0$ and angle DBC= ${30}^0$. Calculate $\angle BDC$

1. $\angle BDC={40}^o$

2. $\angle BDC={70}^o$

3. $\angle BDC={50}^o$

4. $\angle BDC={10}^o$

Q.

Three tangents are drawn at random to a given circle: Show that the odds are $3$ to $1$ against the circle being inscribed in the triangle formed by them.

Q. If the tangent at a point P to a circle with centre O cuts a line through O at Q such that PQ = 24 cm and OQ = 25 cm. Find the radius of the circle.

Q.

Assertion: The measure of $\angle AOC={60}^{\circ }$

Reason: Angle subtended by an arc of a circle at the center of the circle is double the angle subtended by the arc on the circumference.

Which of the following is correct?

1. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion

2. Both Assertion and Reason are true and Reason is the correct explanation of Assertion

3. Assertion is true but Reason is false

4. Assertion is false but Reason is true

Q.

In two concentric circles, prove that all chords of the outer circle, which touch the inner circle, are of equal length. Then, AB = CD

1. True

2. False

Q. Four alternative answers for each of the following questions are given. Choose the correct alternative.
(1) Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between their centers ?
(2) Two circles intersect each other such that each circle passes through the centre of the other. If the distance between their centres is 12, what is the radius of each circle ?
(3) A circle touches all sides of a parallelogram. So the parallelogram must be a, ................... .
(4) Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.
(5) If two circles are touching externally, how many common tangents of them can be drawn?
(6) ∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB).
(7) Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6, EB = 10, CE = 8, find ED.
(8) In a cyclic ▢ABCD, twice the measure of ∠A is thrice the measure of ∠C. Find the measure of ∠C?
(9) Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No point, except point B, is common to the arcs. Which is the type of ∆ABC?

(10) Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ is an acute angle. (ii) ∠XYZ can’t be a right angle. (iii) ∠XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of ∠XYZ.

Q.

In the given figure, $PT$ is a tangent to the circle. If $PA=5cm$ and $PT=10cm,$ then the radius of the circle is ___ $cm.$

1. $4cm$

2. $2cm$

3. $7.5cm$

4. $3cm$

Q.

Maximum number of tangents parallel to a given secant of a circle are four.

1. 1

2. 2

Q.

A tangent at any point on the circle is __ to the radius through the point of contact.

Q. Question 5
Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5 m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6 m each, what is the distance between Reshma and Mandip?

Q.

In two concentric circles of radii 6 cm and 10 cm with centre ‘O’. OP is the radius of the smaller circle, OP ⊥ AB which cuts the outer circle at A and B. Find the length of AB.

Q.

A circle touches the side BC of ∆ ABC at P, AB and AC produced at Q and R respectively, then perimeter of ∆ ABC = ___ AR.

1. 4

2. 1

3. 3

4. 2

Q.

To draw a pair of tangents to a circle which are inclined to each other at an angle of 35°, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is

1. 55°

2. 105°

3. 145°

4. 140°

Q.

Find the area in ${cm}^2$ of the shaded region where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre. [2 MARKS]

Q.

In the figure given A, B, C and D are four points on the circumference of a circle with centre O. Arc AB = 2 arc BC and ∠AOB = 108⁰. Calculate and justify your answer.

1. 46, 20, 106

2. 46, 23, 106

3. 46, 23, 126

4. 54, 25, 106

5. 54, 27, 110

6. 54, 27, 126

Q. In the given figure AB is a chord of a circle with centre O and BT is a tangent. If $\angle ACB={75}^{\circ }$, then $\angle ABT=$ ______.

1. ${105}^{\circ }$
   
2. ${15}^{\circ }$
3. ${70}^{\circ }$
   
4. ${75}^{\circ }$
   

Q.

$\overleftrightarrow{CX}$ is a tangent to a circle with centre A, at a point X. $\overline{AX}$ = 5 cm, CX = 12 cm, AC = ?

1. 12 cm

2. 4 cm

3. 9 cm

4. 13 cm

Q.

In given figure, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, $\angle$B = ${90}^{\circ }$ and DS = 5 cm, then the radius of the circle is ___ cm.

1. 11

2. 18

3. 6

4. 15

Q. Find, in terms of π, the length of the arc that subtends an angle of 30° at the centre of a circle of radius 4 cm.

Q.

In the given figure, O is the centre of the circle and $\angle ABC={55}^{\circ }$. Calculate the values of x and y.

Q.

A circle of radius r is inscribed in a triangle of area ${}^{\prime }{\Delta }^{\prime }$. If the semi-perimeter of the triangle is s, then the correct relation is

1. $2r=\frac{\Delta }{s}$
   

2. $r=\frac{\Delta }{s}$
   

3. $r=\frac{s}{\Delta }$
   

4. $2s=\Delta r$

Q. In the given figure, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA ⊥ PB
then the length of each tangent. [CBSE 2013]
(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm

Q.

In two concentric circles of radii 6 cm and 10 cm with centre ‘O’. OP is the radius of the smaller circle, OP ⊥ AB which cuts the outer circle at A and B. Find the length of AB.

Q.

In the given figure, C and D are points on the semi-circle described on AB as diameter. Given angle BAD = ${70}^0$ and angle DBC= ${30}^0$. Calculate $\angle BDC$

Q.

In the given figure 'O' is the center of the circle. SP and TP are the two tangents at S and T respectively. $\angle SPT$ is ${50}^{\circ }$, the value of $\angle SQT$ is:

1. 125°

2. 65°

3. 115°

4. None of the above