Are these practice questions for Circles based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 10 Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Circles free?

Yes. Every chapter includes free practice questions for Class 10 students to get you started.

What types of questions are included?

The practice set for Circles mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Circles for full coverage.

How many practice questions are there for Circles?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Circles Class 10 Mathematics Practice Questions with Answers

easy

List the number of tangents that can be drawn to a circle from a point based on its position. Identify the three possible positions for the point.

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easy

A quadrilateral PQRS is drawn to circumscribe a circle. If

PQ = 12

cm,

QR = 15

cm, and

RS = 14

cm, calculate the length of SP.

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easy

How many tangents can be drawn to a circle from a point lying inside it?

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easy

Fill in the blank: The tangent at any point of a circle is ________ to the radius through the point of contact.

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easy

Define the following terms related to a circle: (i) Tangent, (ii) Chord, (iii) Point of Contact.

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easy

Two tangents PA and PB are drawn from an external point P to a circle with centre O. If

\angle AOB = 130^\circ

, calculate the measure of

\angle APB

.

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easy

What is the name of the common point between a tangent and a circle?

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easy

State Theorem 10.1 regarding the relationship between a tangent and the radius of a circle. Describe a diagram that illustrates the theorem.

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easy

Two concentric circles have radii of 13 cm and 5 cm. Formulate a method to find the length of a chord of the larger circle that is tangent to the smaller circle and justify your steps.

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easy

Define a secant of a circle.

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easy

A point Q is 26 cm away from the centre of a circle and the length of the tangent drawn from Q to the circle is 24 cm. Calculate the radius of the circle.

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easy

Justify the statement: 'A circle can have an infinite number of tangents.'

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medium

Summarize the key properties of tangents to a circle. List at least five distinct properties or theorems.

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medium

How many tangents, parallel to a given secant, can a circle have at the most?

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medium

Explain the three possible relationships between a line and a circle in a plane. Use a simple description for each case.

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medium

A circle is inscribed in a right-angled triangle whose legs (sides containing the right angle) are 8 cm and 15 cm. Calculate the radius of the inscribed circle.

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medium

A circle is inscribed in a

\triangle ABC

with sides

AB = 12

cm,

BC = 8

cm, and

AC = 10

cm. The circle touches the sides at D, E, and F respectively. Evaluate the lengths of AD, BE, and CF.

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medium

Justify why the center of a circle must lie on the angle bisector of the angle formed by two tangents drawn from an external point.

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medium

State Theorem 10.2 about the lengths of tangents from an external point. Describe a diagram that illustrates this property.

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medium

Evaluate why it is impossible to construct a tangent to a circle from a point located inside it.

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medium

A circle is inscribed in a

\triangle ABC

having sides

AB = 8

cm,

BC = 10

cm, and

AC = 12

cm. The circle touches the sides at points D, E, and F on AB, BC, and AC respectively. Calculate the lengths of AD, BE, and CF.

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medium

A circle with center O has a radius of 5 cm. A point P is located 13 cm from O. Two tangents, PQ and PR, are drawn from P to the circle. Calculate the area of the quadrilateral PQOR.

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medium

Justify that the tangents drawn at the endpoints of a diameter of a circle are parallel.

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medium

Two tangents are drawn to a circle from an external point P, touching the circle at A and B. The circle has a radius of 9 cm and the centre O is 15 cm away from P. Calculate the length of the chord AB.

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medium

Explain the concept of 'length of the tangent' from an external point to a circle. Use a diagram description to show two tangents from an external point P to a circle with centre O and points of contact T₁ and T₂. Identify the lengths of the tangents and state the relationship between them.

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medium

Two concentric circles have radii of 13 cm and 5 cm. Calculate the length of the chord of the larger circle that is tangent to the smaller circle.

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medium

From an external point P, two tangents PA and PB are drawn to a circle with centre O and radius

r

. If

\angle APB = 60^\circ

, analyze the geometry to express the length of the tangent PA in terms of the radius

r

.

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medium

Describe the relationship between a secant and a tangent. Explain how a tangent can be considered a special case of a secant.

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medium

A circle touches the side BC of a

\triangle ABC

at point P, and touches the extended sides AB and AC at points Q and R respectively. Analyze the lengths of the tangents from vertices A, B, and C to demonstrate that the length of the tangent AQ is half the perimeter of

\triangle ABC

.

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medium

Formulate a formal proof for the theorem stating that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

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medium

A circle touches the sides of

\triangle PQR

at points A, B, and C on sides PQ, QR, and RP respectively. If

PQ = 12

cm,

QR = 8

cm, and

RP = 10

cm, calculate the length of PA.

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hard

Formulate a proof for this generalization of Pitot's theorem: 'If a hexagon ABCDEF circumscribes a circle, the sum of the lengths of its alternate sides are equal.' That is, prove

AB + CD + EF = BC + DE + FA

.

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hard

Critique the following statement: 'A secant is a generalized form of a tangent.'

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hard

A circle with center O is inscribed in a quadrilateral ABCD. Formulate a proof that the opposite sides of the quadrilateral subtend supplementary angles at the center of the circle. That is, prove

\angle AOB + \angle COD = 180^\circ

.

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hard

From a point P, two tangents PA and PB are drawn to a circle with center O. If the length of OP is equal to the diameter of the circle, justify that

\triangle APB

is an equilateral triangle.

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hard

Explain with the help of a diagram description why it is not possible to draw a tangent to a circle from a point inside it, but it is possible to draw exactly two tangents from a point outside it.

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hard

Formulate a concise argument to prove that a parallelogram circumscribing a circle must be a rhombus.

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hard

Critique the following flawed proof of Theorem 10.1 (The tangent at any point of a circle is perpendicular to the radius through the point of contact). Identify the logical fallacy.

Flawed Proof: 'Let XY be the tangent at point P to a circle with center O. Join OP. In

\triangle OPX

,

\angle OPX

must be

90^\circ

because the tangent and radius meet. Therefore,

OP \perp XY

.'

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hard

Describe the logical steps used to explain why the tangent at any point of a circle is perpendicular to the radius through the point of contact (Theorem 10.1).

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hard

PQ is a chord of length 24 cm in a circle with a radius of 13 cm. Tangents drawn at points P and Q intersect at an external point T. Calculate the length of the tangent TP.

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hard

In a right-angled triangle, create a proof to show that the radius 'r' of its incircle is given by the formula

r = \frac{a + b - c}{2}

, where a and b are the lengths of the legs and c is the length of the hypotenuse.

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hard

Design a proof to show that if two tangents are drawn to a circle from an external point, then they are equally inclined to the line segment joining the center to that point.

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CirclesClass 10 Mathematics Practice Questions