Relations Using Incircle and Outcircle

Q.

AB is the diameter and AC is a chord of a circle with centre O such that angle $BAC={30}^o$. The tangent to the circle at C intersects AB produced in D. Then,

1. BC = BD

2. BC = 5BD

3. BC = 8BD

4. BC = 9BD

Q.

In figure, AB and CD are two parallel tangents to a circle with center O. ST is tangent segment between the two parallel tangents touching the circle at Q. Show that $\angle SOT={90}^{\circ }$. [4 MARKS]

Q.

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Q.

In the given figure AB = 8 cm. M is the midpoint of AB.

Semi circles are drawn on AB, AM and MB as diameter. A circle with centre ‘O’ touches all three semi circles as shown. Prove that the radius of this circle is .

Q.

Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; then the points P, B, Q and T are not concyclic.

1. False

2. True

Q. Question 3
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of ${80}^{\circ },$ then $\angle POA$ is equal to
(A) ${50}^{\circ }$
(B) ${60}^{\circ }$
(C) ${70}^{\circ }$
(D) ${80}^{\circ }$

Q.

In the following figure, PQ and PR are tangents to the circle, with centre O. If $\angle QPR={60}^{\circ }$. Select the statements that are true,

1. $\angle OQR={30}^{\circ }$

2. $\angle QOR={120}^{\circ }$

3. $\angle OQR={30}^{\circ }$

4. $\angle QSR={60}^{\circ }$

Q.

In fig., there are two concentric circles with center O and of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP [3 MARKS]

Q.

AC and BC are two equal chords of a circle with diameter AB forming a $\Delta ABC$ as shown in the figure. If the radius of the circle is 5 cm. find the length of the equal chords.

Q.

In the adjoining figure the diameter of the larger circle is 10 cm and the smaller circle touches internally the larger circle at P and passes through O, the center of the larger circle. Chord SP cuts the smaller circle at R and OR is equal to 4 cm. What is the length of the chord SP?

1. 12 cm

2. 6 cm

3. 9 cm

4. 8√2
   

Q.

Two circles of radii 13 cm and 5 cm touch each other internally. The distance between their centers will be 8 cm.

1. 18 cm

2. 12 cm

Q.

In the given figure, O is the centre of the circle and AB is a tangent to it at point B. If $\angle BDC={65}^{\circ }$, then find $\angle BAO$

1. ${80}^{\circ }$

2. ${40}^{\circ }$

3. ${120}^{\circ }$

4. ${70}^{\circ }$

Q.

Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure.Then, the angles PAQ and PBQ are supplementary.

1. True

2. False

Q.

In the figure, O is the centre of a circle of radius 9 cm. Semicircle with Diameter OA cuts the circle at D and BC is a tangent through B.

If OA = 25 cm , then BC = ____.

1. 15 cm

2. 12 cm

3. 16 cm

4. 24 cm

Q.

In the figure , O is the centre of a circle of radius 9 cm. Semicircle with diameter OA cuts the circle at D.

If DP is perpendicular to OA and OA = 15 cm, then DP = _____ cm.

1. 8.5 cm

2. 7.2 cm

3. 6.2 cm

4. 7 cm

Q.

In the given fig. the circle touch the lines at X, Y, Z, B, C and D. If PQ = 8 cm, QA = 9 cm and PA = 7 cm, then the length PX, AZ and QY are respectively.

1. 3 cm, 4 cm, 5 cm

2. 3 cm, 5 cm, 4 cm

3. 4 cm, 5 cm, 3 cm

4. 4 cm, 3 cm, 5 cm

Q.
If two tangents PT and PS are drawn to a circle with centre O and radius 'r' from an external point P and OP=2r then $\angle$TOP is ____.

1. ${30}^0$
2. ${60}^0$
3. ${120}^0$
4. ${90}^0$

Q.

Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Then, $\angle CPA=$ _______.

1. $5\angle DPB$.

2. $2\angle DPB$.

3. $8\angle DPB$.

4. $\angle DPB$.

Q.

ABC is a right triangle right angle at B such that BC = 6 cm and AB = 8 cm. The radius of its in-circle is

1. 8 cm

2. 5 cm

3. 3 cm

4. 2 cm

Q. AB is the diameter and AC is a chord of a circle such that $BAC={30}^{\circ }$ . If tangent at C intersects AB produced in D, prove that BC=BD.

Q.

If two tangents inclined at an angle${60}^{\circ }$are drawn to a circle of the radius $3cm$ then the length of each tangent is

Q.

In the given figure, PQ is a chord of length 8cm of a circle of radius 5cm. The tangents at P and Q intersect at a point T. Find the length TP. [4 MARKS]

Q. In the following figure, PQ and PR are tangents to the circle, with centre O. If $\angle QPR={60}^{\circ }$, find the value of $\angle OQR$.

Q.

Draw triangle ABC, in which BC = 8 cm, AC = 7 cm and AB = 6 cm with A, B and C as centre. Draw circles of radii 2.5 cm, 3.5 cm and 4.5 cm respectively and show that these circles touch in pairs

Q.

In the figure P , Q and R are the points where the incircle of $△ABC$ touches the sides :

Using the information given , $\angle QPR$ = ____

1. ${35}^{\circ }$

2. ${50}^{\circ }$

3. ${80}^{\circ }$

4. ${70}^{\circ }$

Q. Four equal circles, each of radius a, touch each other. Show that the area between them is $\frac{6}{7}{a}^2$ (Take π = 22/7).

Q.

In the figure below, O is the centre of the circle.

What is the diameter of the circle?

Q.

State true or false.
Two circles of radii 13 cm and 5 cm touch each other internally. The distance between their centers will be 8 cm.

1. 18 cm

2. 12 cm

Q. Radii of 2 circles touching internally are 5 cm and 2.4 cm. Distance between their centres is:

1. 2.6 cm
2. 5 cm
3. 7.4 cm
4. 4.2 cm

Q. Question 6
In figure AB and CD are common tangents to two circles of equal radii. Prove that AB = CD.