Tangent Perpendicular to Radius at Point of Contact

Q. Question 6
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Q. Question 1
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm
(B) 12 cm
(C) 15 cm
(D) 24.5 cm

Q.

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Q.

In figure AB is a diameter of a circle with centre O and AT is a tangent. If $\angle AOQ={58}^{\circ }$, find $\angle ATQ$.

Q. In the given figure, CD is the tangent at Q and $\angle RQD=x^{\circ },$ then the measure of $\angle RPQ$ is equal to

1. 
   ${90}^{\circ }$
2. 
   $(180-x{)}^{\circ }$
3. 
   $x^{\circ }$
4. 
   $(180+x{)}^{\circ }$

Q. Question 6
Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is ${60}^{\circ }$. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

Q. In figure, O is the centre of the circle and BCD is tangent to it at C. Prove that $\angle BAC+\angle ACD={90}^{\circ }$

Q.

A point P is 10 cm away from the center of a circle. The length of the tangent drawn from P to the circle is 8 cm. The radius of the circle is equal to ____.

1. 5 cm

2. 6 cm

3. 4 cm

4. 3 cm

Q.

In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA = 17 cm, then the length of AC (in cm) is

(a) 9 (b) 15 (c) $\sqrt{353}$ (d) 25

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. $r=\frac{\Delta }{s}$
   
2. $r=\frac{s}{\Delta }$
   
3. $2s=\Delta r$
4. $2r=\frac{\Delta }{s}$
   

Q.

The length of the tangent from a point A to a circle, of radius 3 cm, is 4 cm. Find the distance of A from the center of the circle.

1. $\sqrt{7}$ cm

2. 7 cm

3. 5 cm

4. 25 cm

Q.

In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then the radius of the circle is

(a) 10 cm (b) 12 cm (c) 13 cm (d) 15 cm

Q.

In the following figure, $PQ$ is a chord of a circle with centre $O$ and $PT$ is a tangent. If $\angle$ $QPT$ = ${60}^o$, find $\angle$ $PRQ$.

Q.

In figure, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOR is a diameter. If $\angle POR={130}^{\circ }$ and S is a point on the circle, find $\angle 1+\angle 2$

Q.

PQ is a tangent to a circle with centre O at the point P. If $\Delta$ OPQ is an isosceles triangle, then $\angle$ OQP is equal to

(a) ${30}^o$ (b) ${45}^o$ (c) ${60}^o$ (d) ${90}^o$

Q.

In the figure, 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. Statement 1: A tangent is perpendicular to the radius at the point of contact.
Statement 2: A line from the centre to any other point on the tangent has a length greater than the radius of the circle.

1. Statement 1 is the correct explanation for statement 2.
2. Statement 2 is the correct explanation for Statement 1
3. Statement 1 and Statement 2 are false.
4. Statement 2 is a theorem and Statement 1 is false.

Q.

In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If $\angle$ PAB = ${67}^o$, then the measure of $\angle$ AQB is

(a) ${73}^o$ (b) ${64}^o$ (c) ${53}^o$ (d) ${44}^o$

Q.

In the given figure, O is the centre of a circle, AOC is its diameter such that $\angle$ ACB = ${50}^o$ . If AT is the tangent to the circle at the point A then $\angle$ BAT = ?

(a) ${40}^o$ (b) ${50}^o$ (c) ${60}^o$ (d) ${65}^o$

Q.

In the given figure, O is the centre of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circle respectively. If PA = 10 cm, find the length of PB up to one place of decimal.

Q.

In the given figure, PQ and PR are tangents to a circle with centre A. If $\angle$ QPA = ${27}^o$ then $\angle$ QAR equals

(a) ${63}^o$ (b) ${117}^o$ (c) ${126}^o$ (d) ${153}^o$

Q.

If PA and PB are two tangents to a circle with centre O such that $\angle$ AOB = ${110}^o$ then $\angle$ APB is equal to

(a) ${55}^o$ (b) ${60}^o$ (c) ${70}^o$ (d) ${90}^o$

Q.

Equal circles with centres O and O' touch each other at X. OO' produced to meet a circle with centre O', at A. AC is a tangent to the circle whose centre is O. O' D is perpendicular to AC. Find the value of $\frac{D{O}^{\prime }}{CO}$

Q.

Two concentric circles are of diameters $30cm$ and $18cm$. Find the length of the chord of the larger circle which touches the smaller circle.

Q.

PQ is a tangent to a circle with centre O at the point P. If $△$OPQ is an isosceles triangle such that OP = PQ, then find the measure of $\angle$OQP.

1. ${30}^{\circ }$

2. ${45}^{\circ }$

3. ${60}^{\circ }$

4. ${90}^{\circ }$

Q.

In the given figure, O is the centre of the circle. PA and PB are tangents. Show that AOBP is a cyclic quadrilateral.

Q.

In the given figure, O is the centre of a circle, PQ is a chord and PT is the tangent at P. If $\angle$ POQ = ${70}^o$, then $\angle$ TPQ is equal to

(a) ${35}^o$ (b) ${45}^o$ (c) ${55}^o$ (d) ${70}^o$

Q.

In the given figure, DE and DF are tangents from an external point D to a circle with centre A. If DE = 5 cm and DE $\perp$ DF then the radius fo the circle is

(a) 3 cm (b) 4 cm

(c) 5 cm (d) 6 cm

Q. Question 9
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.

If PA and PB are two tangents to a circle with centre O such that $\angle$ APB = ${80}^o$. Then, $\angle$ AOP = ?

(a) ${40}^o$ (b) ${50}^o$ (c) ${60}^o$ (d) ${70}^o$