Solution of Chapter 9. Circles (NCERT - Exemplar Mathematics Book)

If radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is

In figure, if ∠AOB = 125°, then ∠COD is equal to

In figure, AB is a chord of the circle and AOC is its diameter such that ∠ACB = 50°. If AT is the tangent to the circle at the point A, then ∠BAT is equal to

From a point P which is at a distance of 13 cm from the center 0 of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle is drawn. Then, the area of the quadrilateral PQOR is

At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A, is

In figure, AT is a tangent to the circle with center 0 such that OT = 4 cm and ∠OTA = 30°. Then, AT is equal to

In figure, if 0 is the center of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to

In figure, if PA and PB are tangents to the circle with center O such that _{∠APB = 50°, then ∠OAB is equal to}

If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then the length of each tangent is

In figure, if PQR is the tangent to a circle at Q whose center is 0, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to

If a chord AB subtends an angle of 60° at the center of a circle, then angle between the tangents at A and B is also 60°.

The length of tangent from an external point P on a circle is always greater than the radius of the circle.

The length of tangent from an external point P on a circle with center 0 is always less than OP.

The angle between two tangents to a circle may be 0°.

If angle between two tangents drawn from a point P to a circle of radius a and center O is 90°, then

If angle between two tangents drawn from a point P to a circle of radius a and center 0 is 60°, then

The tangent to the circumcircle of an isosceles ΔABC at A, in which AB = AC, is parallel to BC.

If a number of circles touch a given line segment PQ at a point A, then their centers lie on the perpendicular bisector of PQ.

If a number of circles pass through the end points P and Q of a line segment PO, then their centers lie on the perpendicular bisector of PQ.

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersect AB extends at D, then BC = BD.

Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.

Two tangents PQ and PR are drawn from an external point to a circle with center O. Prove that QORP is a cyclic quadrilateral.

Prove that the center of a circle touching two intersecting lines lies on the angle bisector of the lines.

If from an external point B of a circle with center O, two tangents BC and BD are drawn such that ∠DBC = 120°, prove that BC + BD = BO i.e., BO = 2 BC.

In figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD

In figure, AB and CD are common tangents to two circles of equal radii. Prove that AB = CD.

In figure, common tangents AB and CD to two circles intersect at E. Prove that AB = CD.

A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.

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

Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A.

If a hexagon ABCDEF circumscribe a circle, prove that

AB + CD + EF = BC + DE + FA

Let s denotes the semi-perimeter of a ΔABC in which BC = a, CA = b and AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively. Prove that BD = s – b.

From an external point P, two tangents, PA and PB are drawn to a circle with center 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 perimeter of the triangle PCD.

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

Two circles with centers 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.

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

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.

AB is a diameter and AC is a chord of a circle with center O such that ∠BAC = 30°. The tangent at C intersects extended

AB at a point D. Prove that BC = BD.

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.

In a figure the common tangents, AB and CD to two circles with centers O and O’ intersect at E. Prove that the points O, E and O’ are collinear.

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

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

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

A is a point at a distance 13 cm from the center 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 ΔABC.