CBSE Class 10 Mathematics • Chapter 10
Circles Class 10 Worksheet with Solutions and MCQs
Practise tangent properties, lengths and angle calculations. Revise Theorems 10.1 and 10.2, follow worked examples and check your understanding with MCQs, proof questions and FAQs.
This Circles Class 10 worksheet helps you apply the two main tangent theorems to numerical problems and geometrical proofs. Start by locating the centre, point of contact and external point. Joining the centre to the point of contact often reveals a right triangle that makes the calculation straightforward.
Use the worksheet available on this page alongside these explanations and original practice questions. For Circles Class 10 CBSE revision, learn each theorem with its conditions and state the reason behind every important step. A correct diagram and clear reasoning are as useful as the final answer.
Circles Class 10: Tangent and Secant
A circle consists of all points in a plane at a fixed distance from its centre. A tangent is a line that meets the circle at exactly one point, called the point of contact. A secant meets it at two distinct points.
| Position of the Line | Common Points with the Circle | Description |
|---|---|---|
| Outside the circle without touching it | 0 | Non-intersecting line |
| Touches the circle | 1 | Tangent |
| Passes through the circle | 2 | Secant |
From a point inside a circle, no tangent can be drawn. At a point on the circle, there is one tangent. From an external point, two tangents can be drawn.
Circles Class 10 Theorem 10.1
Radius and Tangent Are Perpendicular
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
If PT is tangent to a circle with centre O at T, then OT ⟂ PT and ∠OTP = 90°.
Therefore, △OTP is right-angled at T. If OP and the radius OT are known, Pythagoras’ theorem can be used to find PT.
Circles Class 10 Theorem 10.2
Tangents from the Same External Point Are Equal
The lengths of the two tangent segments drawn from an external point to a circle are equal.
If PA and PB touch the same circle at A and B, then PA = PB.
The external point must be the same. Tangent segments drawn from different external points need not have equal lengths.
Class 10 Circles Formulas and Useful Results
Let O be the centre, r the radius, P an external point and A and B the contact points of its two tangents.
| Result | Formula or Relationship |
|---|---|
| Radius to the point of contact | OA ⟂ PA and OB ⟂ PB |
| Equal tangent lengths | PA = PB |
| Tangent length | PA = √(OP2 − r2), where OP > r |
| Distance from the centre to the external point | OP = √(r2 + PA2) |
| Angle between two tangents | ∠APB = 180° − ∠AOB, using the non-reflex central angle |
| Angle bisector | OP bisects ∠APB |
Circles Class 10 Questions with Solutions
Example 1: Find a Tangent Length
Question: PT is tangent at T to a circle with centre O. If OP = 13 cm and OT = 5 cm, find PT.
OT ⟂ PT by Theorem 10.1.
In right triangle OTP:
OP2 = OT2 + PT2
132 = 52 + PT2
PT2 = 169 − 25
PT2 = 144
PT = 12 cm
Example 2: Use Equal Tangents to Find x
Question: PA and PB are tangents from P to the same circle. If PA = (3x + 2) cm and PB = (5x − 8) cm, find x and the tangent lengths.
PA = PB by Theorem 10.2.
3x + 2 = 5x − 8
10 = 2x
x = 5
PA = 3(5) + 2
PA = 17 cm
PB = 5(5) − 8
PB = 17 cm
Answer: x = 5 and each tangent is 17 cm long.
Example 3: Find the Angle Between Two Tangents
Question: PA and PB are tangents to a circle with centre O. If the non-reflex ∠AOB = 120°, find ∠APB.
∠OAP = 90° and ∠OBP = 90°.
The angles of quadrilateral OAPB add to 360°.
∠APB = 360° − 90° − 90° − 120°
∠APB = 60°
Example 4: Prove That Two Tangents Are Equal
Given: PA and PB are tangents from P to a circle
with centre O, touching it at A and B.
To prove: PA = PB.
Join OA, OB and OP.
In right triangles OAP and OBP:
∠OAP = ∠OBP = 90° — radius perpendicular to tangent
OA = OB — radii of the same circle
OP = OP — common hypotenuse
Therefore, △OAP ≅ △OBP by RHS congruence.
Hence, PA = PB by corresponding parts of
congruent triangles.
Circles Class 10 Case-Based Question
Two Straight Paths to a Circular Garden
A circular garden has centre O and radius 9 metres. A point P is 15 metres from O. Two straight paths PA and PB touch the garden at A and B.
1. What are ∠OAP and ∠OBP?
Both are 90° because each radius is perpendicular to its tangent.
2. Find PA.
PA2 = OP2 − OA2
PA2 = 225 − 81
PA2 = 144
PA = 12 metres
3. Find PB.
PB = PA by the equal-tangents theorem.
PB = 12 metres
4. Find the combined length of the two paths.
PA + PB = 12 + 12
Total = 24 metres
Circles Class 10 MCQ with Answers
1. What angle does a tangent make with the radius at contact?
A. 30° B. 45° C. 60° D. 90°
Answer: D. 90°. This follows from Theorem 10.1.
2. How many tangents can be drawn from an external point?
A. 0 B. 1 C. 2 D. Infinitely many
Answer: C. Two.
3. If PA and PB are tangents from P and PA = 8 cm, find PB.
A. 4 cm B. 8 cm C. 16 cm D. Cannot be found
Answer: B. 8 cm. Tangents from the same external point have equal lengths.
4. If the angle between two tangents is 70°, find the non-reflex central angle.
A. 70° B. 90° C. 110° D. 140°
Answer: C. 110°. The two angles add to 180°.
Extra Class 10 Circles Questions for Practice
- A tangent PT touches a circle of radius 7 cm. If OP = 25 cm, find PT.
- PA and PB are tangents from P. If PA = 2x + 5 and PB = 4x − 7, find x and the common tangent length.
- The non-reflex angle between radii to two contact points is 135°. Find the angle between the tangents.
- A tangent segment is 15 cm long and the radius is 8 cm. Find the distance from the external point to the centre.
- PA and PB are tangents from P. If ∠PAB = 55°, find ∠APB.
Check the Practice Answers
- PT = 24 cm.
- x = 6; each tangent is 17 units long.
- 45°.
- 17 cm.
- PA = PB, so ∠PBA = 55°. Therefore, ∠APB = 180° − 55° − 55° = 70°.
Using This Worksheet with NCERT Exercises 10.1 and 10.2
For Circles Class 10 Exercise 10.1, revise the meaning of a tangent and its relationship with the radius. For Exercise 10.2, practise equal-tangent problems, angle calculations and proof questions. Follow the exact questions in your NCERT edition.
When using Circles Class 10 Exercise 10.2 solutions, attempt each question before checking the answer. Compare the construction, theorem and calculation, rather than only the final result. Apply the same method to verified previous year questions. The examples on this page are original additional practice.
Common Mistakes to Avoid
- Assuming a line is tangent because it looks tangent in a drawing.
- Placing the right angle at the centre instead of the contact point.
- Using OP as the radius when P is an external point.
- Equating tangent lengths drawn from different external points.
- Adding r2 instead of subtracting it when finding a tangent length.
- Writing a proof without giving geometrical reasons.
FAQs on Circles Class 10 Worksheet
What is a tangent to a circle?
A tangent is a line meeting a circle at exactly one point. That point is called the point of contact. The radius drawn to it is perpendicular to the tangent.
What is the difference between a tangent and a secant?
A tangent meets the circle at one point, while a secant meets it at two distinct points. A chord is the segment joining two points on the circle.
What does Circles Class 10 Theorem 10.1 state?
The tangent at a point on a circle is perpendicular to the radius through that point. If PT touches the circle at T and O is the centre, then ∠OTP = 90°.
What does Circles Class 10 Theorem 10.2 state?
The lengths of the two tangent segments drawn from the same external point to a circle are equal. If PA and PB are those tangents, then PA = PB.
How do I calculate a tangent length?
Join the centre to the contact point and external point. The resulting triangle is right-angled at contact. Use PT = √(OP2 − r2). For OP = 13 cm and r = 5 cm, PT = 12 cm.
Why are two tangents from the same external point equal?
The radii to their contact points form two right triangles. These have equal radii and a common hypotenuse from the centre to the external point. RHS congruence gives equal tangent lengths.
How many tangents does a circle have?
A circle has infinitely many tangents overall, one at each point on its circumference. From a particular external point, only two can be drawn. From an interior point, none can be drawn.
Does the line joining the centre and external point bisect the tangent angle?
Yes. The two right triangles formed by the radii and tangents are congruent. Their corresponding angles at the external point are equal, so this line bisects the angle between the tangents.
Which Circles Class 10 questions should I practise?
Practise tangent lengths, equal-tangent equations, angles between tangents, RHS-based proofs and applications involving figures that touch a circle. State the theorem used in each solution.
Are Areas Related to Circles HOTS questions part of this chapter?
Areas Related to Circles is a separate CBSE chapter covering sectors, segments and related area calculations. The Circles chapter focuses on tangent geometry. Choose practice resources according to the topic being assessed.
How should I use a Circles Class 10 worksheet with solutions?
Attempt the questions first. Check your diagram, theorem choice and working against the solution. Reattempt incorrect questions without looking at the answer.
Where can I find Circles Class 10 notes and a mind map?
Visit the Circles Class 10 Notes and Mind Map on WitKnowLearn. Use it for concept revision before returning to this worksheet.
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