About Circles (Tangents & Secants) — Class 10 ICSE

Study tangent properties, alternate segment theorem, and intersecting chords. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 5). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 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.

What you'll learn in Circles (Tangents & Secants)

• Tangents and Their Properties
• Chord Properties and Intersecting Chords
• Angles in Circles
• Two Circles — Touching and Intersecting

Interactive lesson · about 25 minutes · checkpoint question after every unit

Circles (Tangents & Secants) — solved examples for Class 10 ICSE

Practice questions on Circles (Tangents & Secants)

1. Q1.easy

   $AT$ is a tangent to a circle with centre $O$ at the point of contact $A$. If $OT = 4$ cm and $\angle OTA = 30^\circ$, find the length of $AT$.

   1. A)$2$ cm
   2. B)$2\sqrt{2}$ cm
   3. C)$2\sqrt{3}$ cm
   4. D)$4$ cm
   Show answer

   Answer: $2\sqrt{3}$ cm

   Hint: $OA \perp AT$. In right $\triangle OAT$, use the cosine of $\angle OTA$.

2. Q2.easy

   $O$ is the centre of a circle. $PQ$ is a chord and $PR$ is the tangent at $P$. If $\angle QPR = 50^\circ$, find $\angle POQ$.

   1. A)$50^\circ$
   2. B)$80^\circ$
   3. C)$100^\circ$
   4. D)$130^\circ$
   Show answer

   Answer: $100^\circ$

   Hint: Tangent $\perp$ radius gives $\angle OPQ$. Then $\triangle OPQ$ is isosceles with $OP = OQ$.

3. Q3.easy

   $O$ is the centre of a circle, $AB$ is a chord, and $AT$ is the tangent at $A$. If $\angle AOB = 100^\circ$, find $\angle BAT$.

   1. A)$40^\circ$
   2. B)$50^\circ$
   3. C)$60^\circ$
   4. D)$80^\circ$
   Show answer

   Answer: $50^\circ$

   Hint: $\triangle OAB$ is isosceles. Then $\angle BAT = 90^\circ - \angle OAB$.

4. Q4.medium

   $AP$ and $BP$ are tangents from $P$ to a circle with centre $O$. If $AP = 5$ cm and $\angle APB = 60^\circ$, find the length of chord $AB$.

   1. A)$2.5$ cm
   2. B)$5$ cm
   3. C)$5\sqrt{2}$ cm
   4. D)$5\sqrt{3}$ cm
   Show answer

   Answer: $5$ cm

   Hint: $AP = BP$ makes $\triangle PAB$ isosceles. With $\angle APB = 60^\circ$, the triangle is equilateral.

5. Q5.medium

   $PT$ and $PS$ are tangents from external point $P$ to a circle with centre $O$. If $OP = 2r$ where $r$ is the radius, find $\angle TPS$.

   1. A)$30^\circ$
   2. B)$45^\circ$
   3. C)$60^\circ$
   4. D)$90^\circ$
   Show answer

   Answer: $60^\circ$

   Hint: In right $\triangle OTP$, $OT = r$ and $OP = 2r$, so $\sin(\angle OPT) = 1/2$.

6. Q6.medium

   Two equal circles with centres $O$ and $O'$ touch each other externally at $X$. $AC$ is a tangent from external point $A$ on circle $O'$ to the other circle at $C$. If $O'D \perp AC$ at $D$, find the ratio $\dfrac{O'D}{OC}$.

   1. A)$1:3$
   2. B)$1:2$
   3. C)$2:3$
   4. D)$1:1$
   Show answer

   Answer: $1:3$

   Hint: $O'D \parallel OC$ (both $\perp AC$). Triangles $AO'D$ and $AOC$ are similar — set up the ratio of corresponding sides using $AO' : AO$.

7. Q7.hard

   $\triangle ABC$ is right-angled at $B$ with $AB = 12$ cm and $BC = 5$ cm. Find the radius of the circle inscribed in the triangle.

   1. A)$1$ cm
   2. B)$2$ cm
   3. C)$2.5$ cm
   4. D)$3$ cm
   Show answer

   Answer: $2$ cm

   Hint: For a right triangle with legs $a$, $b$ and hypotenuse $c$, the inradius is $r = (a + b - c)/2$.

8. Q8.hard

   Two circles with centres $O$ and $O'$ of radii $6$ cm and $3$ cm touch each other internally at $A$. A chord $AB$ of the larger circle meets the smaller circle at $C$. If $AB = 10$ cm, find $AC$.

   1. A)$2.5$ cm
   2. B)$3.33$ cm
   3. C)$5$ cm
   4. D)$7.5$ cm
   Show answer

   Answer: $5$ cm

   Hint: Use that $AA'$ (extended diameters) are corresponding parts of similar triangles formed by the chord.

9. Q9.hard

   $O$ is the centre of a circle with radius $5$ cm. From an external point $P$ where $OP = 13$ cm, two tangents $PA$ and $PB$ are drawn. Find the length of $PA$.

   1. A)$8$ cm
   2. B)$10$ cm
   3. C)$12$ cm
   4. D)$\sqrt{194}$ cm
   Show answer

   Answer: $12$ cm

   Hint: $OA \perp PA$, so apply Pythagoras in right $\triangle OAP$.

These are 9 of the 60 questions available for Circles (Tangents & Secants). Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.