In the given figure, O is the center of a circle, PQ is a chord and the tangent PT at P makes an angle of 50° with PQ. Then, ∠POQ = ?

If the angle between two radii of a circle is 130° then the angle between the tangents at the ends of the radii is

If tangents PA and PB from a point P to a circle with center O are drawn so that ∠APB = 80° then ∠POA = ?

In the given figure, AD and AE are the tangents to a circle with center O and BC touches the circle at F. If AE = 5 cm then perimeter of ΔABC is

In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively.

If AB = x cm, BC = 7 cm, CR = 3 cm and

AS = 5 cm, find x.

In the given figure, PA and PB are the tangents to a circle with center O. Show that the points A, 0, B, P are concyclic.

In the given figure, PA and PB are two tangents from an external point P to a circle with center O. If ∠PBA = 65°, find ∠OAB and ∠APB.

Two tangent segments BC and BD are drawn to a circle with center O such that ∠CBD = 120°. Prove that OB = 2BC.

Fill in the blanks.

(i) A line intersecting a circle in two distinct points is called a ……..

(ii) A circle can have ………..parallel tangents at the most.

(iii) The common point of a tangent to a circle and the circle is called the ………..

(iv) A circle can have ………..tangents.

Prove that the lengths of two tangents drawn from an external point to a circle are equal.

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

In the given figure, if AB = AC, prove that BE = CE.

If two tangents are drawn to a circle from an external point, show that they subtend equal angles at the center.

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Prove that the parallelogram circumscribing a circle, is a rhombus.

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

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal.

Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

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 segments joining the points of contact at the center.

PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangents at P and Q intersect at a point T as shown in the figure.

Find the length of TP.