Angle between Tangents Drawn from an External Point

Q. Question 6
In a right angle $\angle$ ABC is which $\angle B={90}^{\circ }$, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at P bisects BC.

Q. Prove that the lengths of two tangents drawn from an external point to a circle are equal.

Q. If AP and AQ are the two tangents to a circle with centre O so that $\angle POQ$ = 120°, then$\angle PAQ$ is equal to ___.

1. 70$°$
2. 60$°$
3. 80$°$
4. 90$°$

Q.

In the given figure, XY is a diameter of the circle, PQ is a tangent to the circle at Y. Given that $\angle$AXB = ${50}^0$ and $\angle$ABX = ${70}^0$, calculate $\angle$APY.s

Q.

$\text{PA and PB are the two tangents drawn}\text{to the circle. O is the center of the circle.}\text{A and B are the points of contact of the}\text{tangents PA and PB with the circle.}\text{If}\angle OPA={35}^{\circ },\text{then find}\angle POB.$

1. ${55}^{\circ }$

2. ${65}^{\circ }$

3. ${75}^{\circ }$

4. ${85}^{\circ }$

Q. In the given figure, PA and PB are two tangents drawn from an external point P. If $\angle APB={40}^{\circ }$, then $\angle PAB=$

1. ${70}^{\circ }$
2. ${90}^{\circ }$
3. ${60}^{\circ }$

Q. Question 4
If AB is a chord of a circle with centre O, AOC is a diameter and tangent T at A as shown in the figure. Prove that $\angle$BAT = $\angle$ACB.

Q. Question 3
From an external point P, two tangents , PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD.

Q. AP and AQ are the two tangents drawn to a circle with centre O, these tangents are inclined to each other at an angle ${60}^0$. Find $\angle APQ$ in the given figure.

1. ${80}^{\circ }$
2. ${70}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q.

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Q. Question 6
In figure, AT is a tangent to the circle with centre O such that OT = 4 cm and $\angle OTA={30}^{\circ }$ . Then , AT is equal to
(A) 4 cm
(B) 2 cm
(C) 2 $\sqrt{3}$ cm
(D) 4 $\sqrt{3}$ cm

Q. In the given figure, AP and AQ are the tangents drawn to a circle from a point A outside the circle. If $\angle$PAQ = ${40}^{\circ }$ then, find $\angle$AQP.

1. ${90}^{\circ }$
2. ${35}^{\circ }$
3. ${70}^{\circ }$
4. ${20}^{\circ }$

Q. In the given figure, O is the centre of the circumcircle. Tangents at A and C intersect at P. Given $\angle$ $AOB={140}^{\circ }$ and $\angle$$APC$ = ${80}^{\circ }$; find the $\angle$ $BAC$.

1. ${90}^{\circ }$
2. ${20}^{\circ }$
3. ${60}^{\circ }$
4. ${50}^{\circ }$

Q. Question 6
In a right angle $\angle$ ABC is which $\angle B={90}^{\circ }$, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at P bisects BC.

Q. Question 7
In figure, tangents PQ and PR are drawn to a circle such that $\angle RPQ={30}^{\circ }$. A chord RS is drawn parallel to the tangent PQ. Find the $\angle$ RQS.

Q.

Two tangents PA and PB are drawn to a circle with center O from an external point P. Prove that $\angle APB=2\angle OAB$ [3 MARKS]

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. In the given figure, AP and AQ are the tangents drawn to a circle from a point A outside the circle. If $\angle$PQA= ${65}^{\circ }$ then, find $\angle$PAQ.

1. ${100}^{\circ }$
2. ${25}^{\circ }$
3. ${90}^{\circ }$
4. ${50}^{\circ }$

Q. Question 4
If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that $\angle$ DBC = ${120}^{\circ }$, prove that BC+BD=BO i.e BO = 2BC.

Q. If $TP$ and $TQ$ are two tangents to a circle with centre $O$ so that $\angle POQ={110}^o$, then $\angle PTQ$ is equal to

1. ${60}^o$
2. ${80}^o$
3. ${70}^o$
4. ${90}^o$

Q. In the given figure, PA and PB are two tangents drawn from an external point P. If $\angle APB={40}^{\circ }$, then $\angle AOB=$

( 2 marks)

Q. Question 4
If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that $\angle$ DBC = ${120}^{\circ }$, prove that BC+BD=BO i.e BO = 2BC.

Q. Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$, then $\angle PTQ=\angle OPQ$.

1. True
2. False

Q.

Two tangents PA and PB are drawn from an external point P 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 = 2 AP

4. OP = AP

Q. In figure, the chord $AB$ of the larger of the two concentric circles, with centre $O,$ touches the smaller circle at $C.$ Prove that $AC=CB.$

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. Draw a circle with the help of a bangle. Take a point $P$ outside the circle. Construct the pair of tangents from this point $P$ to the circle.

Q. In the given figure PQ and PR are two tangents on a circle with centre O. If $\angle QPR={60}^{\circ }$, then $\angle QRO={30}^{\circ }.$

1. True
2. False

Q. Question 6
In figure, AT is a tangent to the circle with centre O such that OT = 4 cm and $\angle OTA={30}^{\circ }$ . Then , AT is equal to
(A) 4 cm
(B) 2 cm
(C) 2 $\sqrt{3}$ cm
(D) 4 $\sqrt{3}$ cm

Q. In the given figure, PA and PC are the tangents. Find the value of the angle APC.

1. 90 degrees
2. 30 degrees
3. 50 degrees
4. 60 degrees