# Circles

**Exercise 1**

**1.** How many tangents can a circle have?

**2.** Fill in the blanks:

(i) A tangent to a circle intersects it in_______ point(s).

(ii) A line intersecting a circle in two points is called a _____.

(iii) A circle can have________ parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called_____.

**3.** A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is:

(A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm

**4. ** Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

**Exercise 2**

In Q.5 to 7, choose the correct option and give justification.

**5.** 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

**6. ** In figure, if TP and TQ are the two tangents to a circle with centre 0 so that ∠ POQ = 110^{0}, then PTQ is equal to:

(A) 60^{0} (B) 70^{0} (C) 80^{0} (D) 90^{0}

**7.** If tangents PA and PB from a point P to a circle with centre 0 are inclined to each other at angle of 80^{0}, then ∠ POA is equal to:

(A) 50^{0} (B) 60^{0} (C) 70^{0} (D) 80^{0}

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

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

**10. ** 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.

**11. ** 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.

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

**13. ** In figure, XY and X’Y’ are two parallel tangents to a circle with centre 0 and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠ AOB = 90^{0}.

**14.**Prove that the angel 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.

**15.** Prove that the angel 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

**16. ** Prove that the parallelogram circumscribing a circle is a rhombus.

**17. ** 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.

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