## Circle Important Questions

**1. If quadrilateral ABCD is drawn to circumscribe a circle then prove that AB+CD=AD+BC.**

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

**3. AB is a chord of length 9.6cm of a circle with centre O and radius 6cm.If the tangents at A and B intersect at point P then find the length PA.**

**4. The incircle of a ∆ABC touches the sides BC, CA &AB at D,E and F respectively. If AB=AC, prove that BD=CD.**

**5. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre of the circle**

**6. PQ and PR are two tangents drawn to a circle with centre O from an external point P. Prove that angleQPR=2angleOQR.**

**7 Prove that the length of tangents drawn from an external point to a circle is equal. Hence, find BC, if a circle is inscribed in a ∆ABC touching AB,BC &CA at P,Q &R respectively, having AB=10cm, AR=7cm &RC=5cm.**

**8. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following: If O is the centre of two concentric circles, AB is a chord of the larger circle touching the smaller circle at C, then prove that AC=BC.**

**9. A circle touches the side BC of a ∆ABC at a point P and touches AB and AC when produced, at Q & R respectively. Show that AQ=1/2 (perimeter of ∆ABC).**

**10.. From an external point P, a tangent PT and a line segment PAB is drawn to circle with centre O, ON is perpendicular to the chord AB. Prove that PA.PB=PN2 -AN2 .**

**11. If AB is a chord of a circle with centre O, AOC is diameter and AT is the tangent at the point A, then prove that angleBAT=angleACB.**

**12. The tangent at a point C of a circle and diameter AB when extended intersect at P. IfanglePCA=1100 , find angleCBA.**

**13. If PA and PB are tangents from an external point P to the circle with centre O, the find AOP+OPA.**

**14. ABC is an isosceles triangle with AB=AC, circumscribed about a circle . Prove that the base is bisected by the point of contact.**

**15. AB is diameter of a circle with centre O. If PA is tangent from an external point P to the circle with POB=1150 then find OPA.**

**16. PQ and PR are tangents from an external point P to a circle with centre . If RPQ=1200 , Prove that OP=2PQ.**

**17. If the common tangents AB and CD to two circles C(O,r) and C’(O’r’) intersect at E, then prove that AB=CD. 6. If a, b, c are the sides of a right triangle where c is the hypotenuse , then prove that radius r of the circle touches the sides of the triangle is given by r= (a+b-c)/2.**

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Sample paper 1

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