Circles — Class 10 solved problems

32 problems from this chapter, each solved step by step.

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
The length of the chord of the larger circle which touches the smaller circle is 8 cm.
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length of PQ.
The length of PQ is √119 cm.
From an external point P, two tangents PA and PB are drawn to a circle with centre O. If angle APB = 70 degrees, find angle AOB.
The measure of AOB is 110^°.
If PA and PB are tangents from an external point P to a circle with centre O such that APB = 70^°, find AOB
The measure of AOB is 110^°.
Find the length of a tangent from a point Q at a distance of 25 cm from the center, given that the radius of the circle is 7 cm (or 24 cm tangent length with 25
The length of the tangent is 24 cm.
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that BD = 8 cm and DC = 6 cm. Find the sides AB and AC.
The sides AB and AC are 15 cm and 13 cm respectively.
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 times angle OPQ.
PTQ = 2 OPQ
Two chords AB and CD of a circle intersect at a point P inside the circle. Prove that AP × PB = CP × PD. State the theorem that justifies your answer to part (2
The proof shows that AP × PB = CP × PD. This is justified by the Theorem of Intersecting Chords.
A circle with centre O has two chords AB and CD such that AB = CD. The chords intersect at a point P inside the circle. It is given that: * AP = 4 cm * PB = 5 c
The problem statement contains inconsistent information because the given segment lengths (AP=4 cm, PB=5 cm, CP=2.5 cm) and the condition AB=CD cannot be simult
PQ is a chord of length 9cm of a circle of radius 6 cm. The tangents to the circle at points P and Q intersect each other at an external point T. Find the lengt
The length of tangent TP is (18√7)/(7) cm.
PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at points P and Q intersect each other at an external point T. Find the leng
The length of tangent TP is (20)/(3) cm.
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
The length of the chord of the larger circle which touches the smaller circle is 8 cm.
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengt
The lengths of the sides are AB = 15 cm and AC = 13 cm.
From an external point P a tangent PT is drawn to a circle of radius 8 cm. If the distance OP from the centre is 17 cm, find the length of the tangent PT.
The length of the tangent PT is 15 cm.
Radius of a circle is 15cm. How far should the center of another circle be from the midpoint of this circle to ensure that there is no overlap in their circumfe
The problem is underdetermined. The distance between the center of the first circle and the center of the second circle must be greater than or equal to 15 + R_
(a) If a hexagon PQRSTU circumscribes a circle, prove that, PQ + RS + TU = QR + ST + UP. OR (b) In the given figure, two concentric circles have radii 3 cm and
The length of TP is 12 cm.
In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If APB = 60^°, then the length of PA is:
The length of PA is 5√3 cm.
A circle is touching the side BC of ABC at P and touching the sides AB and AC produced at Q and R respectively. Prove that AQ = (1)/(2) (AB + BC + AC).
AQ = (1)/(2) (AB + BC + AC)
In the given figure, x, y and z are the sides of a right triangle, where z is the hypotenuse. Prove that the radius r of the circle which touches the sides of t
The radius r of the inscribed circle is r = (x + y - z)/(2).
Prove that the lengths of tangents drawn from an external point to a circle are equal.
The lengths of tangents drawn from an external point to a circle are equal, i.e., PQ = PR.
Assertion (A) TA and TB are two tangents drawn from an external point T to a circle with centre 'O'. If TBA = 75^° then ABO = 25^°. Reason (R) The tangent drawn
Assertion (A) is false, but Reason (R) is true.
A circle is touching the side BC of a ABC at the point P and touching AB and AC produced at points Q and R respectively. Prove that AQ = (1)/(2) (Perimeter of A
AQ = (1)/(2) (Perimeter of ABC)
The length of the tangent drawn from a point P, whose distance from the centre of a circle is 25 cm, and the radius of the circle is 7 cm, is:
24 cm
In the figure, PA and PB are two tangents to the circle with centre O such that APB = 50^°. Then, the measure of OAB is:
The measure of OAB is 25^°.
From a point P which is at a distance of 17 cm from the centre O of a circle of radius 8 cm, the pair of tangents PQ and PR to the circle are drawn. The area of
The area of triangle OPQ is 60 cm².
Case Study - 1 36. In a park, four poles are standing at positions A, B, C and D around the circular fountain such that the cloth joining the poles AB, BC, CD a
(i) OSA = 90^° (ii) ABCD is a Kite (iii) (a) AP = 4 cm (iii) (b) QOR = 120^°
Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. [Figure required: t
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Case Study - 1 36. In a park, four poles are standing at positions A, B, C and D around the circular fountain such that the lines joining the poles AB, BC, CD a
(i) OSA = 90^° (ii) The figure ABCD is a kite. (iii) (a) AP = 4 cm OR (iii) (b) QOR = 120^°
VARIATION: Three circles of radius 1 touch each other externally. Find the area of the triangle formed by joining their centers. Find the numerical value if exa
The area of the triangle formed by joining their centers is √3 square units.
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ = 2 OPQ. [Figure required: circle with centre O, external po
Hence, it is proved that PTQ = 2 OPQ.
केंद्र O वाले वृत्त पर बाह्य बिंदु T से दो स्पर्श रेखाएँ TP तथा TQ खींची गई हैं। सिद्ध कीजिए कि PTQ = 2 OPQ है। [आकृति आवश्यक: केंद्र O वाला वृत्त, बाह्य बिंदु
यह सिद्ध हो गया है कि PTQ = 2 OPQ है।
PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. [Figure: circle centre O radius 5 c
The length TP is (20)/(3) cm.

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