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Circles

The tangent at a point C of a circle and a diameter AB when extended intersect at P. If  ∠PCA =110º , find  ∠CBA

Circles

In Figure. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.

Circles

In Figure, the common tangent, AB and CD to two circles with centres O and O’ intersect at E. Prove that the points O, E, O’ are collinear.

Circles

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.

Circles

AB  is  a  diameter  and AC  is  a  chord  of  a  circle  with  centre  O  such  that ∠BAC = 30°. The tangent at C intersects extended AB at a point D. Prove that BC = BD.

Circles

In Figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.

[Hint: Draw a line through Q and perpendicular to QP.]

Circles

In a right triangle ABC in which  ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC.

Circles

Two  circles  with  centres  O  and  O’  of  radii  3  cm  and  4  cm,  respectively intersect at two points P and Q such that OP and O’P are tangents to the two circles. Find the length of the common chord PQ.

Circles

If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in Figure. Prove that ∠BAT = ∠ACB.

Circles

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the the perimeter of the triangle PCD.