Maharashtra SSC Geometry HOTS Questions with Answers – Circle | Class 10 Board Exam 2027

Asked in: July 2024 board paper (same question with radii 5.5 cm and 4.2 cm in July 2025; radii 5 cm and 3 cm as a 1-mark question in March 2024; circles touching internally in March 2026)

Two circles of radii 5.5 cm and 3.3 cm respectively touch each other externally. Then the distance between their centres is ____

Asked in: March 2025 board paper

In cyclic quadrilateral ABCD, \(m\angle A = 100^\circ\), then find \(m\angle C\).

Asked in: March 2025 and March 2026 board papers

In the following figure chord MN and chord RS intersect at point D. If RD = 15, DS = 4, MD = 8, find DN.

[Figure: a circle with chords MN and RS crossing at point D inside the circle.]

Asked in: July 2025 and March 2026 board papers

In the following figure, seg PS is a tangent segment, line PR is a secant. If PQ = 3.6, QR = 6.4, find PS.

[Figure: from external point P, tangent PS touches the circle at S; secant P – Q – R cuts the circle at Q and R.]

Asked in: March 2024, July 2025 and June 2026 board papers

In the following figure, \(m(\text{arc } NS) = 125^\circ\), \(m(\text{arc } EF) = 37^\circ\). Find the measure of \(\angle NMS\).

[Figure: chords EN and FS of a circle, extended, intersect at point M outside the circle; arc EF is the near arc and arc NS the far arc.]

Asked in: March 2024 board paper

\(\angle ABC\) is inscribed in arc ABC of a circle with centre O. If \(\angle ABC = 60^\circ\), find \(m\angle AOC\).

Activity: \(\angle ABC = \dfrac{1}{2} m(\text{arc AXC})\) ...... ____; \(60^\circ = \dfrac{1}{2} m(\text{arc AXC})\); ____ \(= m(\text{arc AXC})\). But \(m\angle AOC = m(\text{arc } \ldots)\) (property of central angle) \(\therefore m\angle AOC = \) ____

[Figure: circle with centre O; B on the major arc; AXC is the minor arc intercepted by \(\angle ABC\).]

Asked in: July 2024 and June 2026 board papers

Prove that, ‘Opposite angles of a cyclic quadrilateral are supplementary’.

Asked in: March 2025 and March 2026 board papers

Prove that: The tangent segments drawn from an external point to a circle are congruent.

Asked in: July 2025 board paper

In the following figure, point O is the centre of the circle and length of chord AB is equal to the radius of the circle. Find the measures of (i) \(\angle AOB\) (ii) arc AB (iii) \(\angle ACB\), by completing the activity.

Activity: In \(\Delta AOB\), AO = OB = AB \(\therefore \Delta AOB\) is an ____ triangle. \(\therefore m\angle AOB = \) ____. \(m\angle AOB = m(\text{arc AB}) = \) ____ (definition of measure of an arc). \(m\angle ACB = \dfrac{1}{2} \times \) ____ (inscribed angle theorem) \(= \dfrac{1}{2} \times 60^\circ\) \(\therefore m\angle ACB = \) ____

[Figure: circle with centre O; chord AB equal to the radius; C is a point on the major arc.]

Asked in: July 2025 board paper

In the following figure \(\Delta ABC\) is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and AC are congruent. A circle touches the three sides of triangle as shown. Find the length of tangent segment from A to circle.

[Figure: the incircle of \(\Delta ABC\) touches AB at P, BC at Q and AC at R.]

Asked in: July 2024 board paper

The diameter of circular garden is 13 metre. The distance between two gates of garden is 13 metre. An electric pole is to be erected on the circumference of the garden such that the difference between distance of the pole from each gate is 7 metre. Can such pole be erected? If yes, find the distance of the pole from each gate.

Asked in: March 2024 and June 2026 board papers

AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A. Write answers of the following questions: (a) Draw the figure using given information. (b) Find the measures of \(\angle CAT\) and \(\angle ABC\) with reasons. (c) Whether \(\angle CAT\) and \(\angle ABC\) are congruent? Justify your answer.

Two circles of radii 7 cm and 4 cm touch each other internally. The distance between their centres is

An inscribed angle intercepts an arc of measure 140°. The measure of the inscribed angle is

How many common tangents can be drawn to two circles that touch each other externally?

In a circle with centre O, seg PQ is a diameter and R is a point on the circle. If \(\angle PQR = 35^\circ\), find \(\angle QPR\).

A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent segment from P to the circle.

\(\square PQRS\) is a cyclic quadrilateral. If \(\angle P = (2x + 10)^\circ\) and \(\angle R = (x + 35)^\circ\), find x, \(\angle P\) and \(\angle R\).

Chords AB and CD of a circle intersect at point E inside the circle. If \(m(\text{arc } AC) = 80^\circ\) and \(m(\text{arc } BD) = 50^\circ\), find \(m\angle AEC\) and \(m\angle CEB\).

[Figure: chords AB and CD cross at E; arc AC and arc BD are the arcs opposite each other, intercepted by \(\angle AEC\) and \(\angle BED\).]

Line \(l\) touches a circle at point A. Chord AB makes an angle of 65° with line \(l\). C is a point on the circle on the other side of chord AB from this angle. Find \(m\angle ACB\) and the measure of the arc AB intercepted by the 65° angle.

Prove that, ‘The measure of an inscribed angle is half the measure of the arc intercepted by it’.

Prove that a cyclic parallelogram is a rectangle.

The incircle of \(\Delta ABC\) touches sides AB, BC and CA at points P, Q and R respectively. If AB = 12 cm, BC = 8 cm and CA = 10 cm, find AP, BQ and CR.

Prove the tangent secant segments theorem: If a secant through an external point P meets a circle at B and C (P – B – C), and PA is a tangent segment from P touching the circle at A, then \(PA^2 = PB \times PC\).

Two circles touch each other externally at point P. Seg AB is a common tangent touching the circles at A and B. The common tangent at P meets AB at Q. Prove that (i) Q is the midpoint of AB (ii) \(\angle APB = 90^\circ\).

A circular park has a diameter of 25 m. Two gates A and B are at the two ends of a diameter. A water tap is to be fixed on the boundary of the park such that the difference of its distances from the two gates is 17 m. Is it possible? If yes, find the distance of the tap from each gate.