Forum

Construction

To divide a line segment AB in the ratio p : q (p, q are positive integers), draw a ray AX so that ∠BAX is an acute angle and then mark points on ray AX at equal distances such that the minimum number of these points is

(A) greater of p and q (B) p + q (C) p + q – 1 (D) pq Solution : Answer (B)

Circles

A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP  and AQ  are  the  tangents  to  the  circle  at  P  and  Q.  If  a  tangent  BC  is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ∆ABC.

Circles

 If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle.

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.