**Circles**

**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

**Solution:**

**Question 2.In figure, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to**

(a) 60°

(b) 70°

(c) 80°

(d) 90°

**Solution:**

**Question 3.If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to**

(a) 50°

(b) 60°

(c) 70°

(d) 80°

**Solution:**

**Question 4.Prove that the tangents drawn at the ends of a diameter of a circle are parallel.**

**Solution:**

**Question 5.Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.**

**Solution:**

**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.**

**Solution:**

**Question 7.Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.**

**Solution:**

**Question 8.A quadrilateral ABCD is drawn to circumscribe a circle (see figure). Prove that AB + CD = AD + BC.**

**Solution:**

**Question 9.In figure, XY and X’Y’ are two parallel tangents to a circle , x with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°.**

**Solution:**

**Question 10.Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.**

**Solution:**

**Question 11.Prove that the parallelogram circumscribing a circle is a rhombus.**

**Solution:**

**Question 12.A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see figure). Find the sides AB and AC.**

**Solution:**

**Question 13.Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.**

**Solution:**