CH – 10 CIRCLES
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Understand basic circle properties, master geometric theorems step by step, and build confidence to score maximum marks in your CBSE Class 10 Board Exam!
Chapter Overview
In this chapter, you will learn how lines and circles interact with each other in a flat plane:
- Tangent to a circle: Understanding what a tangent line is and how it touches a circle.
- Point of contact: Locating the exact single point where a tangent touches the circle.
- Radius and tangent: Understanding how the radius line connects to the tangent line.
- The relationship between a tangent and radius: Proving that the radius drawn through the point of contact is perpendicular (90°) to the tangent.
- Tangents from an external point: Finding out how many tangents can be drawn from a point outside the circle.
- Important theorems based on tangents: Learning how to prove and apply key geometry theorems in board exam questions.
🎯 Board Exam Focus
Practise the following core skills carefully to prepare well for your board exam:
- Understanding what a tangent line is when looking at any geometric figure.
- Identifying the point of contact clearly on a circle.
- Understanding the radius and tangent relationship to find missing angles.
- Applying the theorem about the radius being perpendicular to the tangent at the point of contact.
- Understanding how tangents are drawn from an external point.
- Applying the equal tangents theorem to solve side lengths in triangles and quadrilaterals.
- Solving simple and multi-step geometry problems step by step using these concepts.
Most Important Concepts
What is a Circle?
A circle is a closed round shape drawn on a flat paper. It is made of all the points that are at the exact same distance from a fixed point inside it.
- Centre: The fixed point right in the middle of the circle. We usually name it O.
- Radius: The straight line distance from the centre O to any point on the circle. All radii of a circle are equal in length.
- Diameter: A straight line passing through the centre O, connecting two points on the circle. The diameter is always double the radius (Diameter = 2 × Radius).
Plaintext
.---. Legend / Labels:
/ | \ ----------------
/ | r \ O = Centre
| A--O--B | r = Radius (OC)
\ | / AB = Diameter (A to B through O)
\ | /
'-C-' Diameter = 2 × Radius
What is a Tangent?
A tangent to a circle is a straight line that touches the circle at exactly one point.
- Point of Contact: The single common point where the tangent line touches the circle.
Plaintext
.---.
/ \
| O | O = Centre of Circle
\ / XY = Tangent Line
'-P-' P = Point of Contact
X ------------•------------ Y (Touches at exactly ONE point)
- Difference between a Tangent and a Secant:
- A secant is a line that cuts through the circle and crosses it at two points.
- A tangent only touches the circle at one point and does not cut inside the circle.
- A non-intersecting line stays completely outside and does not touch or cut the circle at all.
Plaintext
[ TANGENT ] [ SECANT ]
.---. .---.
/ \ / A \ Line cuts circle
| O | | --•----|-- B at TWO points:
\ / \ / A and B
'-P-' '---'
X ---------•--------- Y
(Touches at 1 point: P)
🧠 Tangent and Radius
There is an important angle rule between the radius and the tangent of a circle:
The radius drawn through the point of contact is always perpendicular to the tangent line.
- Perpendicular means that two lines meet each other at a square corner.
- The angle formed between the radius and the tangent line at the point of contact is always 90°.
- If line XY is a tangent at point P, and O is the centre, then OP is perpendicular to XY (Angle OPT = 90°).
Plaintext
.---.
/ \
| O | O = Centre
\ | r / OP = Radius
' | ' XY = Tangent Line
X -------------P------------- Y ∠OPT = 90°
|
└───▶ [ OP ⊥ XY ] Radius ⊥ Tangent at Point of Contact P
Converse of the Tangent Theorem
The converse of the tangent theorem turns the main rule around:
If a line is drawn through an end point of a radius and forms a 90° angle (is perpendicular) to that radius, then that line must be a tangent to the circle.
This helps us prove that a given line is a tangent when we already know the 90° angle.
Plaintext
.---.
/ \
| O | GIVEN: Radius OP meets Line AB at P
\ | / such that OP ⊥ AB (Angle = 90°)
' | '
A -------------P------------- B CONCLUSION: Line AB MUST be a Tangent
| to the circle at point P.
└───▶ [ ∠OPA = 90° ⇒ AB is a Tangent ]
Tangents from an External Point
An external point is a point that lies outside the circle.
- From a point inside a circle, you can draw 0 tangents.
- From a point on the circle, you can draw exactly 1 tangent.
- From an external point outside the circle, you can draw exactly 2 tangents to the circle.
When you draw two tangents from an external point P to touch the circle at points A and B:
- Point P is the external point.
- Points A and B are the points of contact.
- PA and PB are the two tangents.
Plaintext
.--- A ---. P = External Point
/ | \ PA = Tangent 1 (Point of Contact A)
/ | \ PB = Tangent 2 (Point of Contact B)
P ------------- O | O = Centre
\ \ | /
\ \ | / THEOREM RESULT:
------------------- B ---' PA = PB (Lengths are EQUAL)
🧠 Equal Tangents Theorem
The lengths of two tangents drawn from the same external point to a circle are always equal.
If two tangents PA and PB are drawn from an external point P to a circle with centre O:
- PA = PB
Simple Example:
If a point P is outside a circle, and the length of tangent PA is 8 cm, then the length of the second tangent PB from point P will also be 8 cm.
How to Solve Tangent Questions
Follow these clear steps to solve geometry questions in this chapter:
Plaintext
A (Point of Contact)
/|
/ | r (Radius)
(Tangent) / |
/ |
/ 90°|
(External Pt) P ─────O (Centre)
\ |
\ | r
\ |
\ |
\|
B (Point of Contact)
RIGHT-ANGLED TRIANGLE ΔOAP:
• Angle OAP = 90° (Radius ⊥ Tangent)
• Hypotenuse = OP (Distance from Centre to External Point)
• Base / Height = OA (Radius 'r') and AP (Tangent Length)
• PYTHAGORAS THEOREM: (OP)² = (OA)² + (AP)²
- Look carefully at the diagram: Read the question and trace every line in the figure.
- Identify the centre and radius: Locate centre O and mark all radius lines.
- Identify the tangent line: Find the line that touches the circle.
- Locate the point of contact: Find the point where the radius and tangent meet.
- Mark the 90° angle: Put a 90° symbol at the point of contact between the radius and tangent.
- Apply Pythagoras Theorem: In right-angled triangle OAP:(Hypotenuse)² = (Base)² + (Height)²(OP)² = (Radius)² + (Tangent Length)²
- Use equal tangent lengths: If two tangents come from the same outer point, mark them as equal (PA = PB).
- Solve step by step: Write down statement reasons clearly (like “Radius is perpendicular to tangent”) to get full marks in exams.
📌 Important Theorems & Results
- Definition of Tangent: A tangent touches a circle at exactly one single point.
- Theorem 10.1: The radius through the point of contact is perpendicular to the tangent (Angle = 90°).
- Converse Theorem:A line drawn perpendicular to a radius at its outer end point on the circle is a tangent.
- Theorem 10.2: Tangents drawn from the same external point to a circle are equal in length.
- Angle Bisector Property: The line joining the centre to an external point bisects the angle between the two tangents.
- Parallel Tangents: A circle can have at most two parallel tangents at the ends of a diameter.
🧠 How to Know Which Theorem to Use?
Here is how you can decide which geometric rule to use when solving questions:
- Use the 90° Radius-Tangent Rule when:
- You see a right-angled triangle formed by the radius, tangent, and centre line.
- You need to calculate a missing side using Pythagoras Theorem.
- You need to find unknown internal angles of a triangle or quadrilateral.
- Use the Equal Tangents Rule (PA = PB) when:
- A circle is inscribed inside a triangle or quadrilateral.
- Tangents are drawn from common corner vertices.
- You need to find boundary lengths or perimeter of shapes around a circle.
- Use the Converse Theorem when:
- The question asks you to “Prove that line XY is a tangent to the circle.”
🧠 Concepts Students Find Difficult
- Identifying a tangent correctly: Confusing a line segment inside a circle with a tangent touching on the outside.
- Finding the point of contact: Missing the exact point where the 90° right angle is created.
- Understanding why radius makes a 90° angle: Forgetting that the shortest distance from centre to a line is always perpendicular.
- Understanding the difference between tangent and secant: Forgetting that secants cut through two points while tangents touch one point.
- Identifying equal tangents: Not noticing which external point created the set of tangents.
- Reading geometry diagrams carefully: Mixing up radius length with the line segment extending to an outer point.
⚠️ Common Mistakes Students Make
- Confusing a tangent with a secant:
- Mistake: Drawing a line through the circle across two points and calling it a tangent.
- Fix: Remember that a tangent stays outside and touches the circle at only one point.
- Forgetting the 90° angle at the point of contact:
- Mistake: Forgetting to use 90° when a radius meets a tangent.
- Fix: Whenever you see a radius meeting a tangent, immediately draw a 90° square symbol at that point.
- Mixing up hypotenuse in Pythagoras calculations:
- Mistake: Taking the tangent line as the hypotenuse instead of the line joining centre to external point.
- Fix: The hypotenuse is always the line opposite the 90° angle, which is the line connecting centre O to external point P.
- Using equal tangent lengths for different outer points:
- Mistake: Assuming two tangent lines are equal even when they come from different outer points.
- Fix: Check carefully that both tangents originate from the exact same external point.
- Assuming lines are equal without using a theorem:
- Mistake: Writing PA = PB directly without giving a reason.
- Fix: Always write the reason in brackets: (Lengths of tangents from an external point are equal).
- Misreading distance from centre versus distance from circle surface:
- Mistake: Taking “distance of point from circle” as total distance from centre.
- Fix: Read carefully. Distance from centre = Radius + Distance from circle surface.
🔥 Priority Revision
⭐⭐⭐ HIGH-PRIORITY PRACTICE
- Proving Theorem 10.2: “Lengths of tangents drawn from an external point to a circle are equal”.
- Questions on concentric circles where a chord of the larger circle touches the smaller circle.
- Quadrilaterals circumscribing a circle (proving AB + CD = AD + BC).
⭐⭐ IMPORTANT PRACTICE
- Right-angled triangle calculations finding radius or tangent length using Pythagoras theorem.
- Tangents drawn at the ends of a diameter are parallel.
- Angle calculations involving central angle and angle between two tangents.
⭐ ADDITIONAL REVISION
- Basic fill-in-the-blanks and 1-mark multiple choice questions on secants, tangents, and points of contact.
- Basic conceptual drawing questions.
⚡ Circles Quick Revision Sheet
- Tangent: A straight line touching a circle at exactly 1 point.
- Secant: A straight line intersecting a circle at 2 points.
- Point of Contact: The point where the tangent touches the circle.
- 90° Relationship: Radius is perpendicular to the tangent at the point of contact (OP ⊥ XY).
- Tangents from External Point: Exactly 2 tangents can be drawn from an outer point.
- Equal Tangents: Tangent lengths from the same outer point are equal (PA = PB).
- Important Formula: (OP)² = (Radius)² + (Tangent Length)².
📝 Test Yourself
- A line touches a circle at point P. The centre of the circle is O. If the radius is 7 cm and line segment OP is drawn, what is the angle between the radius and the line at point P?
- From an external point A, the length of tangent AB to a circle with centre O is 12 cm. If the distance of point A from centre O is 13 cm, find the radius of the circle.
- A point P is at a distance of 25 cm from the centre O of a circle. If the radius of the circle is 7 cm, calculate the length of the tangent drawn from point P to the circle.
- Two concentric circles have radii of 10 cm and 6 cm. Find the length of the chord of the larger circle which touches the smaller circle.
- Tangents PA and PB are drawn from an external point P to a circle with centre O. If angle APB = 70°, find the measure of angle AOB.
- Two tangents PA and PB are drawn from a point P to a circle with centre O such that angle AOB = 120°. Find the measure of angle OAP and angle APB.
- Prove that the tangents drawn at the ends of a diameter of a circle are parallel to each other.
- A quadrilateral ABCD is drawn to circumscribe a circle touching its sides at points P, Q, R, and S. If AB = 6 cm, BC = 7 cm, and CD = 4 cm, find the length of side AD.
- From a point T outside a circle with centre O, two tangents TP and TQ are drawn. Prove that angle PTQ = 2 × angle OPQ.
- A circle touches all four sides of a quadrilateral ABCD. Prove that the sum of opposite sides AB + CD is equal to the sum of opposite sides AD + BC.
🎯 Before Your Exam
☐ I understand what a tangent is and how it differs from a secant. ☐ I can identify the point of contact on a circle figure. ☐ I know that the radius is perpendicular to the tangent at the point of contact. ☐ I remember to use 90° angles to solve right-angled triangle questions. ☐ I understand the converse of the tangent theorem. ☐ I know that two tangents drawn from the same external point are equal in length. ☐ I can choose the correct theorem for simple and multi-step geometry questions.
Keep practising your diagrams and step-by-step proofs, and you will achieve great results in your Mathematics exam!