From a point Q, thelength of the tangent to a circle is 24 cm and the distance of Q from thecentre is 25 cm.

**Answer
1** :

The radius of the circle is

(A) 7 cm (B) 12 cm

(C) 15 cm (D) 24.5 cm

Answer:

First, draw a perpendicular from the center Oof the triangle to a point P on the circle which is touching the tangent. Thisline will be perpendicular to the tangent of the circle.

So, OP is perpendicular to PQ i.e. OP ⊥ PQ

From the above figure, it is also seen that △OPQ is a right angled triangle.

It is given that

OQ = 25 cm and PQ = 24 cm

By using Pythagoras theorem in △OPQ,

OQ^{2} = OP^{2} +PQ^{2}

(25)^{2 }= OP^{2}+(24)^{2}

OP^{2} = 625-576

OP^{2} = 49

OP = 7 cm

So, option A i.e. 7 cm is the radius of thegiven circle.

In Fig. 10.11, if TPand TQ are the two tangents to a circle with centre O so that ∠POQ = 110°,

**Answer
2** :

then ∠PTQ is equal to

(A) 60° (B) 70°

(C) 80° (D) 90°

Answer:

From the question, it is clear that OP is theradius of the circle to the tangent PT and OQ is the radius to the tangents TQ.

So, OP ⊥ PT and TQ ⊥ OQ

∴∠OPT = ∠OQT = 90°

Now, in the quadrilateral POQT, we know thatthe sum of the interior angles is 360°

So, ∠PTQ+∠POQ+∠OPT+∠OQT = 360°

Now, by putting the respective values we get,

∠PTQ +90°+110°+90° = 360°

∠PTQ = 70°

So, ∠PTQ is 70° which isoption B.

**Answer
3** :

then ∠ POA is equal to

(A) 50° (B) 60°

(C) 70° (D) 80°

Solution

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

**Answer
4** :

Prove that theperpendicular at the point of contact to the tangent to a circle passes throughthe center.

**Answer
5** :

The length of atangent from a point A at distance 5 cm from the centre of the circle is 4 cm.Find the radius of the circle.

**Answer
6** :

Two concentriccircles are of radii 5 cm and 3 cm. Find the length of the chord of the largercircle which touches the smaller circle.

**Answer
7** :

A quadrilateral ABCDis drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD +BC

**Answer
8** :

**Answer
9** :

Prove that the anglebetween the two tangents drawn from an external point to a circle issupplementary to the angle subtended by the line-segment joining the points ofcontact at the center.

**Answer
10** :

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