How many practice questions are available for Circles?

SparkEd Math offers 60+ practice questions for Circles across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Circles content free on SparkEd Math?

Yes! All Circles content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Circles differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Circles problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Circles content aligned with the Olympiad syllabus?

Yes, all Circles questions and lessons on SparkEd Math are fully aligned with the Mathematical Olympiad Preparation (Olympiad) syllabus for Class 10. The content follows the official curriculum structure and exam pattern.

Circles — Class 10 Olympiad (2025-26)

About Circles — Class 10 Olympiad

Prove and apply tangent properties, solve problems on tangent from an external point, and chord-tangent relations. This topic is part of the Olympiad Class 10 mathematics syllabus (chapter: Module 9). On this page you can practice 50 questions across three difficulty levels — 20 easy, 11 medium, and 19 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Circles — solved examples for Class 10 Olympiad

Example 1easy

A tangent PT touches a circle with center O at point T. If the radius of the circle is 5 cm and the length of the tangent PT is 12 cm, what is the distance from the external point P to the center O?

A)13 cm
B)10 cm
C)17 cm
D)√119 cm

Step-by-step solution

The radius OT is perpendicular to the tangent PT at the point of contact T. Therefore, ΔOTP is a right-angled triangle with the right angle at T.
Applying the Pythagorean theorem in ΔOTP:
$OP² = OT² + PT²$
Given OT (radius) = 5 cm and PT (tangent length) = 12 cm, substitute these values:
$OP² = 5² + 12² = 25 + 144 = 169$
Therefore, the distance from P to O is:
$OP = √169 = 13 cm$

Answer: 13 cm

Example 2medium

From an external point P, two tangents PA and PB are drawn to a circle with center O. A third tangent is drawn at point C on the circle, which intersects PA at X and PB at Y. If ∠APB = 70°, what is the measure of ∠XOY?

A)35°
B)45°
C)55°
D)65°

Step-by-step solution

In quadrilateral OAPB, ∠OAP = ∠OBP = 90° (radius is perpendicular to tangent at point of contact). The sum of angles in a quadrilateral is 360°.
So, ∠AOB + ∠APB = 180°. Given ∠APB = 70°, we have ∠AOB = 180° - 70° = 110°.
Since OX and OY are angle bisectors of ∠AOC and ∠BOC respectively (tangents from X to A and C, and from Y to B and C), ∠XOC = 1/2 ∠AOC and ∠YOC = 1/2 ∠BOC.
Therefore, ∠XOY = ∠XOC + ∠YOC = 1/2 (∠AOC + ∠BOC) = 1/2 ∠AOB. Substituting ∠AOB = 110°, we get ∠XOY = 1/2 × 110° = 55°.

Answer: 55°

Example 3hard

A triangle ABC is circumscribed about a circle with radius 4 cm. The points of tangency on sides AB, BC, CA are D, E, F respectively. If the lengths of segments AF, BD, CE are in arithmetic progression, and the perimeter of triangle ABC is 48 cm, what is the area of triangle ABC?

A)72 cm²
B)84 cm²
C)96 cm²
D)108 cm²

Step-by-step solution

Let AF = x, BD = y, CE = z. Due to tangent properties, AE = x, BF = y, CD = z. The side lengths are AB = x+y, BC = y+z, CA = z+x.
The semi-perimeter s = (AB+BC+CA)/2 = (2x+2y+2z)/2 = x+y+z. Given perimeter = 48 cm, so s = 24 cm. Thus, x+y+z = 24.
The segments AF, BD, CE are in arithmetic progression, so x, y, z are in AP. This means 2y = x + z.
Substitute x+z = 2y into the semi-perimeter equation: 2y + y = 24, which gives 3y = 24, so y = 8 cm. (We don't actually need x or z for the area calculation).
The area of a triangle circumscribing a circle is given by Area = r × s, where r is the inradius. Given r = 4 cm and s = 24 cm. Area = 4 × 24 = 96 cm².

Answer: 96 cm²

Practice questions on Circles

Q1.easy
A quadrilateral ABCD is drawn to circumscribe a circle. If AB = 6 cm, BC = 7 cm, and CD = 6 cm, what is the length of AD?
1. A)5 cm
2. B)6 cm
3. C)7 cm
4. D)8 cm
Show answer
Answer: 5 cm
Hint: Recall the property of tangents drawn from an external point to a circle, specifically how it relates to the sides of a circumscribing quadrilateral.
Q2.easy
From an external point P, two tangents PA and PB are drawn to a circle with center O. If ∠APB = 60°, what is the measure of ∠AOB?
1. A)60°
2. B)90°
3. C)120°
4. D)150°
Show answer
Answer: 120°
Hint: Consider the quadrilateral formed by O, A, P, B. What are the angles at A and B due to the tangent-radius property?
Q3.easy
Two concentric circles have radii 6 cm and 10 cm. A tangent to the inner circle is a chord of the outer circle. What is the length of this chord?
1. A)8 cm
2. B)10 cm
3. C)16 cm
4. D)20 cm
Show answer
Answer: 16 cm
Hint: Draw a diagram. The radius of the inner circle is perpendicular to the tangent at the point of contact. This tangent becomes a chord of the outer circle.
Q4.medium
In a circle with center O, a tangent PT is drawn at point T. A chord TA is drawn. If ∠PTA = 35°, and B is a point on the circle such that AB is a diameter, find the measure of ∠TOB.
1. A)100°
2. B)110°
3. C)120°
4. D)130°
Show answer
Answer: 110°
Hint: Use the Alternate Segment Theorem to relate the angle between the tangent and chord to an inscribed angle. Then, consider the properties of radii in a circle, especially when forming an isosceles triangle.
Q5.medium
Two circles with radii 9 cm and 4 cm touch externally. A direct common tangent PQ is drawn, where P and Q are points of contact. Find the length of PQ.
1. A)12 cm
2. B)10 cm
3. C)8 cm
4. D)6 cm
Show answer
Answer: 12 cm
Hint: Draw a line through the center of the smaller circle parallel to the common tangent. This will form a rectangle and a right-angled triangle, allowing the use of the Pythagorean theorem.
Q6.medium
A quadrilateral ABCD is drawn to circumscribe a circle. If the ratio AB : BC = 2 : 3, the ratio CD : DA = 5 : 4, and the perimeter of the quadrilateral is 36 cm, find the length of the shortest side.
1. A)6 cm
2. B)36/5 cm
3. C)5 cm
4. D)36/7 cm
Show answer
Answer: 36/7 cm
Hint: Recall the property of a quadrilateral circumscribing a circle: the sum of opposite sides are equal. Express the sides in terms of common ratios and solve.
Q7.hard
Two circles with radii R and r (R > r) touch externally. A direct common tangent touches them at P and Q. A third circle is drawn such that it is tangent to both given circles and to the common tangent PQ. What is the radius of this third circle?
1. A)Rr / (R+r)
2. B)(R+r) / (2√Rr)
3. C)(R+r - 2√Rr) / 4
4. D)Rr / (√R+√r)²
Show answer
Answer: Rr / (√R+√r)²
Hint: Consider the relationship between the radii of three circles that are mutually tangent and also tangent to a common line. A well-known formula connects their radii.
Q8.hard
From an external point P, tangents PA and PB are drawn to a circle with center O. A chord AC is drawn parallel to PB. If ∠APB = 70°, find ∠AOC.
1. A)110°
2. B)100°
3. C)120°
4. D)140°
Show answer
Answer: 110°
Hint: Use properties of equal tangents, isosceles triangles, parallel lines, and the relationship between central and inscribed angles.
Q9.hard
From a point P outside a circle, a tangent PT and a secant PAB are drawn, where T is the point of tangency and A, B are points on the circle. If PT = 12 cm and PA = 6 cm, find the length of the chord AB.
1. A)12 cm
2. B)18 cm
3. C)24 cm
4. D)6 cm
Show answer
Answer: 18 cm
Hint: Apply the Tangent-Secant Theorem (also known as Power of a Point Theorem).

These are 9 of the 50 questions available for Circles. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.