**CIRCLES - COMPREHENSIVE CHEAT SHEET**
**1. BASIC DEFINITIONS & RELATIONSHIPS BETWEEN LINES AND CIRCLES**
• Circle: Collection of all points in a plane at constant distance (radius) from a fixed point (centre)
• Non-intersecting line: Line and circle have NO common points
• Secant: Line intersecting circle at exactly TWO points
• Tangent: Line intersecting circle at exactly ONE point (point of contact)
• The tangent is a special case of secant where both endpoints of the chord coincide
**2. THEOREM 10.1 - TANGENT PERPENDICULARITY (MOST IMPORTANT)**
Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Key Insight: If P is point of contact and O is centre, then OP ⊥ tangent at P
Proof Summary:
Critical Corollary: At any point on a circle, there is ONE and ONLY ONE tangent
**3. NUMBER OF TANGENTS FROM DIFFERENT POSITIONS**
**Case 1 - Point INSIDE Circle:**
• Number of tangents: ZERO (0)
• Reason: All lines through interior point intersect circle at 2 points (all are secants)
• Application: No tangent possible from interior
**Case 2 - Point ON Circle:**
• Number of tangents: EXACTLY ONE (1)
• This is the tangent at that point on the circle
• Perpendicular to radius at that point
**Case 3 - Point OUTSIDE Circle:**
• Number of tangents: EXACTLY TWO (2)
• Both tangents make equal angles with line joining external point to centre
• Both tangents are equidistant from centre
• Points of contact denoted T₁ and T₂
**4. KEY RESULTS FOR PARALLEL TANGENTS**
• Maximum parallel tangents to a circle: TWO (2)
• These two tangents are on opposite sides of the circle
• If a secant is drawn, exactly 2 tangents can be drawn parallel to it (one on each side)
• As parallel lines move toward secant, chord length decreases until it becomes zero (tangent position)
**5. STANDARD PROBLEM TYPES & SOLUTIONS**
**Type A: Finding Length of Tangent from External Point**
Given: Circle with centre O, radius r; external point P at distance d from O; tangent from P touches at T
Solution: Use right-angle triangle OTP (since OT ⊥ PT by Theorem 10.1)
Example: If r = 5 cm and OP = 12 cm, then PT = √(144-25) = √119 cm
**Type B: Identifying Tangent vs Secant vs Non-intersecting Line**
Method: Count intersection points
**Type C: Proving Perpendicularity**
Method: Use Theorem 10.1 directly
**6. COMMON MISTAKES TO AVOID**
✗ Mistake 1: Confusing secant with tangent — Remember: tangent touches at 1 point ONLY
✗ Mistake 2: Forgetting perpendicularity — Tangent is ALWAYS perpendicular to radius at contact point
✗ Mistake 3: Wrong tangent count from external point — Always 2 tangents from outside, not 1
✗ Mistake 4: Calculating tangent length without using Pythagorean theorem — Must use right angle property
✗ Mistake 5: Assuming tangent exists at interior points — NO tangent can be drawn from inside circle
**7. IMPORTANT OBSERVATIONS**
• Real-world example: Wheels on ground — ground line is tangent to wheel (circular cross-section)
• Pulley system: Rope on both sides appears as tangents to pulley wheel
• Parallel tangents: Can only be 2 (one on each side of circle)
• Tangent length from external point P: Both tangents from P to circle are EQUAL in length
• Normal to circle: The radius at point of contact is called the normal (perpendicular to tangent)
**8. EXERCISE QUICK ANSWERS**
• Total tangents to a circle: INFINITE (one at each point on circle)
• Tangent intersects circle in: ONE point
• Line with 2 intersection points: SECANT
• Maximum parallel tangents: TWO (2)
• Common point of tangent and circle: POINT OF CONTACT
**9. FORMULAS TO REMEMBER**
Tangent length from external point P to circle with centre O and radius r:
Distance from centre to tangent = radius (this is what perpendicularity means geometrically)
Q1. A rope wrapped around a pulley makes contact at only one point. Based on Theorem 10.1, why is the rope tangent to the pulley rather than a secant?
Answer: A — A tangent is defined as a line intersecting a circle at exactly one point (Activity 1), and Theorem 10.1 proves the radius at the point of contact is perpendicular to it; option B confuses the definition without connecting to the perpendicularity property that makes this configuration stable.
Q2. A student claims: 'If I draw infinitely many parallel lines, at least three of them will be tangents to the same circle.' Is this claim justified by the chapter content?
Answer: B — Activity 2 explicitly demonstrates that when parallel lines are drawn to a secant, the chord lengths decrease until they become zero on both sides, giving exactly two tangents parallel to the secant, not three or more.
Q3. Assertion (A): At any point on a circle, exactly one tangent can be drawn. Reason (R): The tangent at any point of a circle is perpendicular to the radius through the point of contact. Choose the correct option:
Answer: A — Theorem 10.1 (R) proves the perpendicularity property, which logically enforces that only one tangent can exist at a point—if two distinct tangents existed at P, both would be perpendicular to OP, making them the same line, contradicting the assumption they are distinct.
Q4. Assertion (A): A line inside a circle cannot be a tangent to the circle. Reason (R): Any point on a line inside the circle will either lie inside the circle or on the circle, so the line will always intersect the circle in at least two points. Choose the correct option:
Answer: A — A tangent must have exactly one common point with the circle; if a line lies entirely or partially inside, it cannot have exactly one intersection point, so (A) is true; (R) correctly explains why this is impossible using the definition of tangency from the chapter.
Q5. Consider the bicycle wheel in Fig. 10.4. As the wheel rolls forward, the point of contact with the ground constantly changes. Which property ensures that the tangent line (ground) is always perpendicular to the radius at the instantaneous point of contact?
Answer: A — Theorem 10.1 establishes that any tangent to a circle (the ground is tangent to the wheel at each contact point) is perpendicular to the radius at that point; the wheel's circular geometry combined with this theorem ensures the property holds regardless of the wheel's motion parameters.
Q6. A student observes that a tangent never crosses the circle. Based on the chapter's definition and Activity 1, why is this observation correct?
Answer: A — The definition of a tangent (one point of intersection) combined with Theorem 10.1 (perpendicularity to the radius) geometrically prevents the tangent from entering the circle's interior; option B mischaracterizes the tangent as curved when it is always a straight line.
Q7. Two tangents PQ and RS are drawn at points P and S on a circle. If PQ ∥ RS, what can be concluded about the positions of P and S?
Answer: A — From Activity 2 and Theorem 10.1, if two tangents at different points are parallel, they must be perpendicular to parallel radii; parallel radii occur only when they lie on the same line (same diameter), making P and S diametrically opposite.
Q8. Assertion (A): From a point outside a circle, more tangents can be drawn than from a point inside the circle. Reason (R): A point inside the circle lies on no tangent, whereas a point outside the circle can lie on multiple tangents. Choose the correct option:
Answer: A — (A) is true based on Activity 3 setup; (R) correctly explains it—from inside no tangent exists, from outside tangents can exist because the point is outside the region swept by all radius-perpendiculars.
Q9. A line touches a circle at exactly one point P. The distance from the circle's centre O to any other point on this line is greater than the distance OP. Which property from the chapter best explains this geometric fact?
Answer: A — The perpendicularity established by Theorem 10.1 makes OP the shortest distance from O to the line (perpendicular is the shortest distance), explaining why all other distances are greater; option B is about the circle itself, not the tangent-line relationship.
Q10. Assertion (A): If PQ is a secant and we draw lines parallel to PQ, eventually both result in tangents, one on each side of PQ. Reason (R): As we move a line parallel to a secant, the chord length decreases and eventually becomes zero, creating a tangent when the two endpoints of the chord coincide. Choose the correct option:
Answer: A — Activity 2 demonstrates exactly this process: as parallel lines approach the secant PQ from both sides, chord lengths shrink to zero, leaving one tangent on each side; (R) explains the mechanism behind (A) using the tangent-as-limiting-secant concept.
What is a tangent to a circle?
A line that intersects the circle at exactly one point, called the point of contact.
State Theorem 10.1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
How many tangents can be drawn at a point on a circle?
Exactly one tangent can be drawn at any point on the circle.
What is the difference between a tangent and a secant?
A tangent intersects the circle at one point; a secant intersects the circle at two points.
How many tangents can be drawn from a point inside the circle?
No tangents can be drawn from any point lying inside the circle.
How many tangents can be drawn from a point outside the circle?
Exactly two tangents can be drawn from any point lying outside the circle.
What is the relationship between a tangent and the normal to a circle?
The normal at a point is the line containing the radius through that point; it is perpendicular to the tangent.
In Fig. 10.5, why is Q outside the circle?
If Q were inside the circle, the line XY would intersect the circle at two points and would be a secant, not a tangent.
How many parallel tangents can a circle have at most?
A circle can have at most two parallel tangents on opposite sides of the circle.
What does 'point of contact' mean?
The point of contact is the unique point where a tangent line touches the circle.
Define a tangent to a circle and state Theorem 10.1. [2 marks]
A tangent intersects the circle at one point. Theorem 10.1: tangent is perpendicular to the radius at the point of contact.
From a point P outside a circle with centre O and radius 5 cm, two tangents are drawn touching the circle at points A and B. If OP = 13 cm, find the length of tangent PA. [3 marks]
Use Theorem 10.1 to establish OA ⊥ PA. In right triangle OAP, apply Pythagoras theorem: PA² = OP² − OA² = 169 − 25 = 144, so PA = 12 cm.
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact (Theorem 10.1). Use the property that OP is the shortest distance from centre O to the tangent line. [5 marks]
Let XY be tangent at P, and Q be any point on XY other than P. Show Q lies outside the circle (if inside, XY would be a secant). Prove OQ > OP for all such Q, meaning OP is the perpendicular distance. Conclude OP ⊥ XY using the shortest distance principle.
Practice with interactive flashcards, mind maps, upload your own chapters and get AI study kits instantly
Try StudyOS Free →