NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 9: Circles

If a chord \(AB\) subtends an angle of \(60^\circ\) at the centre of a circle, then the angle between the tangents at \(A\) and \(B\) is also \(60^\circ\). State True/False and justify.

The length of a tangent from an external point on a circle is always greater than the radius of the circle. State True/False and justify.

The length of the tangent from an external point \(P\) on a circle with centre \(O\) is always less than \(OP\). State True/False and justify.

The angle between two tangents to a circle may be \(0^\circ\). State True/False and justify.

If the angle between two tangents drawn from a point \(P\) to a circle of radius \(a\) and centre \(O\) is \(90^\circ\), then \(OP = a\sqrt{2}\). State True/False and justify.

If the angle between two tangents drawn from a point \(P\) to a circle of radius \(a\) and centre \(O\) is \(60^\circ\), then \(OP = a\sqrt{3}\). State True/False and justify.

The tangent to the circumcircle of an isosceles triangle \(\triangle ABC\) at \(A\) (where \(AB=AC\)) is parallel to \(BC\). State True/False and justify.

If a number of circles touch a given line segment \(PQ\) at a point \(A\), then their centres lie on the perpendicular bisector of \(PQ\). State True/False and justify.

If a number of circles pass through the end points \(P\) and \(Q\) of a line segment \(PQ\), then their centres lie on the perpendicular bisector of \(PQ\). State True/False and justify.

\(AB\) is a diameter of a circle and \(AC\) is a chord such that \(\angle BAC = 30^\circ\). If the tangent at \(C\) meets \(AB\) produced at \(D\), then \(BC = BD\). State True/False and justify.

Out of 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 \(P\) to a circle with centre \(O\). Prove that \(QORP\) is a cyclic quadrilateral.

From an external point \(B\) of a circle with centre \(O\), two tangents \(BC\) and \(BD\) are drawn such that \(\angle DBC=120^\circ\). Prove that \(BC+BD=BO\) (equivalently, \(BO=2\,BC\)).

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

In Fig. 9.13, \(AB\) and \(CD\) are common tangents to two circles of unequal radii. Prove that \(AB=CD\).

In Question 5 above, if the radii of the two circles are equal, prove that \(AB=CD\).

In Fig. 9.14, 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 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 every chord that is parallel to the tangent at \(A\).

If a hexagon ABCDEF circumscribes a circle, prove that

\(AB + CD + EF = BC + DE + FA\).

Let \(s\) denote the semi-perimeter of a triangle ABC in which \(BC = a\), \(CA = b\), \(AB = c\). 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 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 perimeter of the triangle PCD.

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 Fig. 9.17. Prove that

\(\angle BAT = \angle ACB\).

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.

In a right triangle ABC in which \(\angle B = 90^\circ\), 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.

In Fig. 9.18, tangents PQ and PR are drawn to a circle such that \(\angle RPQ = 30^\circ\). A chord RS is drawn parallel to the tangent PQ. Find the \(\angle RQS\).

AB is a diameter and AC is a chord of a circle with centre O such that \(\angle BAC = 30^\circ\). 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 Fig. 9.19, 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.

In Fig. 9.20, 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.

The tangent at a point C of a circle and a diameter AB when extended intersect at P. If \(\angle PCA = 110^\circ\), find \(\angle CBA\).

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.

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 \(\triangle ABC\).