Angles in Alternate Segment

Q.

A circle is enclosed by a square such that the sides of the square are tangents to the circle. If the radius of the circle is �r��, and the length of the square is ��l��, find the relation between r and l.

1. r = 2l

2. r = l/2

3. r = l/4

4. r = l

Q.

In the figure, AC is tangent to the circle at O and CB is is the perpendicular from C to O. If the radius of the circle is r, then, select the statement that is true.

1. $r^2=OB\times OA$

2. $8r^2=OB\times OA$

3. $7r^2=OB\times OA$

4. $r^3=OB\times OA$

Q. The value of PA in the figure given below is .

1. 4 cm
2. 2 cm
3. 13 cm
4. 8 cm

Q.

In the given figure O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC =7.5 cm, find the radius of the circle.

1. 22.5 cm

2. 12.5 cm

3. 11.25 cm

4. 11 cm

Q.

In the given figure, the circles touch the lines at X , Y , Z , B , C and D.

If PQ = 8cm , QA = 9cm and PA = 7cm, then the length PX , AZ and QY respectively are :

1. 4 cm ; 3 cm ; 5 cm

2. 3 cm ; 5 cm ; 4 cm

3. 4 cm ; 5 cm ; 3 cm

4. 3 cm ; 4 cm ; 5 cm

Q.

In the figure, the tangent from A touches the circle at B. BC is diameter. The line AC cuts the circle at P.

The tangent at P cuts AB at M. Which of the following options are correct?

1. M is the midpoint of AB.

2. $\angle PAM$ = $\angle PBM$

3. $\angle PAM$ = $\angle APM$

4. $\angle BPM$ = $\angle PBM$

Q. In the given figure $△ABC$ is isosceles with AB = AC and is inscribed in a circle. If DAE is a tangent to the circle, then which of the following statement is true?

1. $△ABC$ is equilateral
2. DE intersects BC
   
3. DE // AB
4. DE // BC
   

Q. In the above figure, O is the centre of the circle, FD is a tangent at C.Then, which of the following statements are true?

1. $\angle ABC={90}^{\circ }$
2. $\angle ACB={30}^{\circ }$
3. $\angle ACD={90}^{\circ }$
4. $\angle ABC={110}^{\circ }$

Q.

In the given figure chord AB extended meets tangent DC at C. If AB = 7cm and BC =9 cm and area of $△CBD=18c{m}^2$,

then area of $△CAD=$ ______

1. $16c{m}^2$
   

2. $24c{m}^2$
   

3. $32c{m}^2$
   

4. $64c{m}^2$

Q.

In the given figure. If ∠ACE = 43⁰ & ∠CAF = 62⁰ then find the value of b.

1. 13^o

2. 15^o

3. 17^o

4. 19^o

5. 21^o

Q.

In the above figure, O is the centre of the circle, FD is a tangent at C.Then, which of the following statements are true?

1. $\angle ABC={90}^{\circ }$

2. $\angle ACB={30}^{\circ }$

3. $\angle ACD={90}^{\circ }$

4. $\angle ABC={110}^{\circ }$

Q.

In the given figure, PQ is a diameter and O is the center of the circle.

If $\angle PRT=\angle QTR={90}^{\circ }$

then which of the following statements are correct?

1. $△PSR\sim △SQT$

2. $\angle PSR=\angle OSQ$

3. $\angle PSR=\angle SPO$

4. $△OPS\sim △OQS$

Q. Find ${\left(x+y\right)}^2$, where ∠SQR = 20° and ∠QSP = 30°. [1 mark]

1. 18000
2. 130
3. 16900
4. 2500

Q. What is the value of angle BAC if the value of angle DBC is y?

1. 0.5y
2. 2y
3. 3y
4. y

Q.

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

Q. Find the value of PA in the figure given below.

Q.

In the given figure, the tangent from A touches the circle at B. BC is a diameter.

The Line AC cuts the circle at P. The tangent at P cuts AB at M. If $\angle PCB$ = $x^{\circ }$ , then $\angle BPM$ = ____.

1. ${90}^{\circ }$ - $x^{\circ }$

2. $2x^{\circ }$

3. $x^{\circ }$

4. $\frac{x^{\circ }}{2}$

Q.

If PA and PB are tangents to a circle with centre O and angle AOB = 100°, then angle APB =_____.

1. 80°

2. 50°

3. 90°

4. 100°

Q. In the given figure AB is a chord of a circle with centre O and BT is a tangent. If $\angle ABT={70}^{\circ }$, then $\angle ACB=$ ______.

1. ${35}^{\circ }$
2. ${55}^{\circ }$
3. ${70}^{\circ }$
4. ${145}^{\circ }$

Q. Find $\angle CDE$
[2 marks]

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.

If PA and PB are tangents to a circle with centre O and angle APB = 90°, then AOBP is a _____.

1. parallelogram

2. rectangle

3. square

4. rhombus

Q.

If $\angle A=30°$ and $\angle B=60°$ in $∆ABC$, then measure $\angle C$.

Q. To draw a pair of tangents to a circle which are inclined to each other at an angle of ${60}^0$, it is required to draw tangents at endpoints of those two radii of the circle, the angle between them should be

1. ${135}^0$
2. ${90}^0$
3. ${60}^0$
4. ${120}^0$

Q. In the given figure $△ABC$ is isosceles with AB = AC and is inscribed in a circle. If DAE is a tangent to the circle, then which of the following statement is true?