Important Circle Theorem Problems
Circle theorem problems are among the most important exam-oriented questions in Class 10 Mathematics. These problems test your understanding of tangent properties, chord-tangent relationships, and the ability to construct proofs.
This topic covers frequently asked problems involving tangents, secants, and circle theorems — the kind that appear as 3-mark and 4-mark questions in board exams.
What is Important Circle Theorem Problems?
Key theorems used:
- Theorem 1: A tangent to a circle is perpendicular to the radius at the point of contact.
- Theorem 2: Tangents drawn from an external point to a circle are equal in length.
- Theorem 3: The angle between a tangent and a chord equals the angle in the alternate segment.
- Theorem 4: If two tangents are drawn from an external point, the line from the point to the centre bisects the angle between the tangents.
Important Circle Theorem Problems Formula
Key formulas and properties:
- If PA and PB are tangents from external point P: PA = PB
- ∠OAP = ∠OBP = 90° (radius ⊥ tangent)
- OP bisects ∠APB
- If AB is a chord and PT is a tangent at T: ∠PTB = ∠TAB (alternate segment theorem)
- In concentric circles, if a chord of the larger circle is tangent to the smaller: the chord is bisected at the point of tangency.
Solved Examples
Example 1: Example 1: Prove tangent is perpendicular to radius
Problem: Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof:
- Let O be the centre, T the point of tangency, and l the tangent at T.
- Take any point P on l (other than T).
- P lies outside the circle (tangent touches at only one point), so OP > OT.
- This means OT is the shortest distance from O to the line l.
- The shortest distance from a point to a line is the perpendicular distance.
- Therefore OT ⊥ l. ∎
Example 2: Example 2: Equal tangents from external point
Problem: From an external point P, tangents PA and PB are drawn to a circle with centre O. Prove PA = PB.
Proof:
- In △OAP and △OBP:
- OA = OB (radii)
- ∠OAP = ∠OBP = 90° (radius ⊥ tangent)
- OP = OP (common side)
- By RHS congruence: △OAP ≅ △OBP
- Therefore PA = PB (CPCT). ∎
Example 3: Example 3: Finding tangent length
Problem: A circle has centre O and radius 5 cm. Point P is 13 cm from O. Find the length of the tangent from P to the circle.
Solution:
- Let PA be the tangent, A on the circle. OA = 5, OP = 13, ∠OAP = 90°.
- PA² = OP² − OA² = 169 − 25 = 144
- PA = 12 cm
Answer: Tangent length = 12 cm.
Example 4: Example 4: Angle between tangents
Problem: From point P, tangents PA and PB are drawn to a circle with centre O. If ∠APB = 60°, find ∠AOB.
Solution:
- In quadrilateral OAPB: ∠OAP + ∠APB + ∠PBO + ∠AOB = 360°
- 90° + 60° + 90° + ∠AOB = 360°
- ∠AOB = 120°
Answer: ∠AOB = 120°.
Example 5: Example 5: Tangent from external point — perimeter
Problem: PA and PB are tangents to a circle from P. A chord AB is drawn. If PA = 8 cm, find the perimeter of △PAB.
Solution:
PA = PB = 8 cm (tangents from external point).
The perimeter depends on AB. Without AB or the angle, we can say:
- Perimeter = PA + PB + AB = 16 + AB
If ∠APB = 60°: △PAB is isosceles with PA = PB = 8 and ∠APB = 60°, making it equilateral. AB = 8.
Perimeter = 8 + 8 + 8 = 24 cm.
Answer: If ∠APB = 60°, perimeter = 24 cm.
Example 6: Example 6: Concentric circles — chord tangent
Problem: Two concentric circles have radii 5 cm and 3 cm. Find the length of a chord of the larger circle that is tangent to the smaller circle.
Solution:
- Let O be the centre, the chord AB of the larger circle is tangent to the smaller circle at T.
- OT ⊥ AB (radius ⊥ tangent). OT = 3 cm.
- OA = 5 cm (radius of larger circle).
- AT = √(OA² − OT²) = √(25 − 9) = √16 = 4 cm
- AB = 2 × AT = 8 cm (perpendicular from centre bisects chord)
Answer: Chord length = 8 cm.
Example 7: Example 7: Tangent to circumscribed triangle
Problem: A circle is inscribed in △ABC with AB = 8, BC = 10, CA = 12. Find the lengths of the tangent segments from each vertex.
Solution:
Let tangent lengths from A = x, B = y, C = z.
- x + y = AB = 8
- y + z = BC = 10
- z + x = CA = 12
Adding all three: 2(x + y + z) = 30 → x + y + z = 15.
- z = 15 − 8 = 7
- x = 15 − 10 = 5
- y = 15 − 12 = 3
Answer: From A: 5 cm, from B: 3 cm, from C: 7 cm.
Example 8: Example 8: Prove OP bisects angle
Problem: PA and PB are tangents from P to a circle with centre O. Prove that OP bisects ∠APB.
Proof:
- In △OAP and △OBP:
- OA = OB (radii), OP = OP (common), PA = PB (equal tangents)
- By SSS: △OAP ≅ △OBP
- ∠APO = ∠BPO (CPCT)
- Therefore OP bisects ∠APB. ∎
Example 9: Example 9: Tangent at end of diameter
Problem: AB is a diameter of a circle. A tangent is drawn at B. A line from A meets the tangent at P and the circle at C. If ∠BAP = 30°, find ∠ABC.
Solution:
- AB is diameter → ∠ACB = 90° (angle in semicircle)
- In △ABC: ∠BAC = 30°, ∠ACB = 90°
- ∠ABC = 180° − 30° − 90° = 60°
Answer: ∠ABC = 60°.
Example 10: Example 10: Two tangent circles
Problem: A quadrilateral ABCD is drawn circumscribing a circle. Prove that AB + CD = BC + DA.
Proof:
Let the circle touch AB at P, BC at Q, CD at R, DA at S.
- AP = AS, BP = BQ, CQ = CR, DR = DS (tangents from external points)
- AB + CD = (AP + PB) + (CR + RD) = (AS + BQ) + (CQ + DS)
- BC + DA = (BQ + QC) + (DS + SA) = (BQ + CQ) + (DS + AS)
- Both expressions are the same sum: AS + BQ + CQ + DS. ∎
Real-World Applications
Why these problems matter:
- Board exams: Tangent proofs and calculations carry 3-4 marks each.
- Competitive exams: Circle theorems appear in Olympiads and entrance tests.
- Engineering: Tangent-circle relationships are used in gear design, cam mechanisms.
- Architecture: Arches, domes, and rounded structures use circle properties.
Key Points to Remember
- Tangent ⊥ radius at the point of contact.
- Two tangents from an external point are equal.
- ∠AOB + ∠APB = 180° (where PA, PB are tangents).
- OP bisects ∠APB (when PA, PB are tangents from P).
- For a circumscribed quadrilateral: AB + CD = BC + DA.
- For concentric circles: chord tangent to inner circle is bisected at the point of tangency.
- Tangent length from external point: √(d² − r²) where d = distance to centre.
- Practice proof writing — stating the theorem, showing congruence, and concluding.
Practice Problems
- A circle has radius 8 cm. A point P is 17 cm from the centre. Find the tangent length.
- PA and PB are tangents from P. If ∠APB = 50°, find ∠AOB.
- Two concentric circles have radii 13 and 5 cm. Find the chord of the outer circle tangent to the inner.
- A circle is inscribed in a triangle with sides 6, 8, 10. Find the tangent lengths from each vertex.
- Prove that in a circumscribed quadrilateral, the sum of opposite sides are equal.
- From an external point 25 cm from the centre of a circle of radius 7 cm, two tangents are drawn. Find the angle between them.
- Prove that the tangent at the end of a diameter is parallel to the tangent at the point diametrically opposite.
Frequently Asked Questions
Q1. What is the most important circle theorem for Class 10?
The two most important are: (1) Tangent is perpendicular to the radius, and (2) Tangents from an external point are equal. Most problems use one or both of these.
Q2. How do you find the tangent length from an external point?
Use Pythagoras: tangent length = √(distance² − radius²). The tangent, radius, and line from centre to external point form a right triangle.
Q3. What is the alternate segment theorem?
The angle between a tangent and a chord at the point of tangency equals the angle in the alternate segment of the circle.
Q4. What is a circumscribed quadrilateral?
A quadrilateral that has a circle inscribed inside it (touching all four sides). For such quadrilaterals, the sum of opposite sides are equal.
Q5. How do I write a good geometric proof?
State what is given, what is to prove, draw a clear figure, identify the relevant theorem/congruence rule, show steps logically, and end with 'Hence Proved' or the QED symbol.
Q6. Can a tangent pass through the centre?
No. A tangent touches the circle at exactly one point and is perpendicular to the radius. It cannot pass through the centre.
Related Topics
- Tangent to a Circle
- Tangents from an External Point
- Tangent is Perpendicular to Radius
- Angle Between Two Tangents
- Circle Theorems Introduction
- Angle Subtended by a Chord
- Equal Chords and Equal Angles
- Perpendicular from Centre to Chord
- Angles in the Same Segment
- Angle in a Semicircle
- Cyclic Quadrilateral
- Number of Tangents from a Point
- Tangent-Secant Relationship
- Properties of Tangents - Summary
