Circles (Class 10): Notes और MCQ Practice

आसान भाषा में

Circle एक ऐसा round figure है जिसके सभी points center से बराबर दूरी पर होते हैं। इस chapter में हमारा main hero है Tangent (स्पर्श रेखा)—एक ऐसी straight line जो circle को सिर्फ एक single point पर touch करके निकल जाती है। Board exam के लिए बस 2 theorems और Pythagoras theorem का खेल है!

Key Concepts & Theorems

• Tangent to a Circle: Line जो circle को exactly 1 point पर intersect करती है। इस common point को Point of Contact कहते हैं।
• Number of Tangents:
• Circle के अंदर के point से: 0
• Circle के ऊपर स्थित point से: 1
• Circle के बाहर के external point से: Exactly 2
• Theorem 10.1: Radius, point of contact पर tangent के साथ 90° का angle बनाती है (Radius ⊥ Tangent).
• Theorem 10.2: किसी external point से circle पर खींचे गए दोनों tangents की length equal होती है (PA = PB).
• Right Triangle Relation: Center O, external point P, aur point of contact A के लिए:

OP² = OA² + AP² (जहाँ OP hypotenuse है).

2 Memory Tricks

1. T-R 90 Rule: Tangent और Radius मिलकर capital 'T' जैसा joint बनाते हैं, यानी इनके बीच का angle हमेशा 90° होगा।
2. 'V' for Victory & Value: External point से बनने वाले दो tangents 'V' shape बनाते हैं—'V' ki dono sides hamesha length me equal hoti hain (PA = PB).

2 Common Board Exam Mistakes

1. Hypotenuse की गलत पहचान: Triangle OAP में 90° angle point of contact (A) पर होता है। इसलिए हमेशा OP ही hypotenuse होगा, tangent AP नहीं!
2. Reason न लिखना: Geometry problems solve करते समय bracket में Theorem का statement (जैसे: Tangents from an external point are equal) ना लिखने पर 0.5 से 1 mark कट जाता है।

Chapter 10: Circles

1. Chapter in Short

In earlier classes, you learned that a circle is a collection of all points in a plane that are at a constant distance (radius) from a fixed point (centre). In Class 10, the focus shifts to the interaction between a circle and a straight line in the same plane.

This chapter introduces the concept of a tangent to a circle, explores its existence, and proves two fundamental theorems:

1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. The lengths of tangents drawn from an external point to a circle are equal.

Mastering these two theorems and their geometric applications makes this one of the most scoring chapters in the Class 10 board exam.

2. Key Concepts Explained Simply

A. A Line and a Circle

When a line and a circle lie in the same plane, only three cases can happen:

1. Non-intersecting Line: The line does not touch or cut the circle at all.
2. Secant: The line cuts through the circle at two distinct points. Think of cutting an apple into two pieces with a knife.
3. Tangent: The line touches the circle at exactly one point.

• Everyday Example: When a bicycle rolls on a flat road, at any instant, each wheel touches the road at exactly one point. The level road acts as a tangent to the circular wheel.

B. Point of Contact

The single common point where a tangent touches the circle is called the point of contact.

C. Key Theorems

• Theorem 10.1 (Radius is perpendicular to Tangent):

If you draw a line from the centre to the point of contact, this radius is perpendicular (at 90°) to the tangent line.

• Why? The shortest distance from a point (the centre) to a straight line (the tangent) is always the perpendicular distance. Every other point on the tangent lies strictly outside the circle, so the point of contact is the closest point to the centre.

• Theorem 10.2 (Tangents from an External Point):

From a point outside a circle, you can draw exactly two tangents. The lengths of these two tangent segments (from the outside point to their respective points of contact) are completely equal.

3. Important Definitions, Facts, and Properties

• Tangent to a Circle: A straight line that touches the circle at only one single point.
• Secant: A straight line that intersects the circle at two distinct points.
• Point of Contact: The point where the tangent touches the circle.
• Number of Tangents from Different Points:
• From a point inside the circle: 0 tangents can be drawn.
• From a point on the circle: Exactly 1 tangent can be drawn.
• From a point outside the circle: Exactly 2 tangents can be drawn.
• Parallel Tangents Fact: A circle can have a maximum of 2 parallel tangents at opposite ends of a diameter.
• Theorem 1 (Perpendicularity): Radius ⊥ Tangent at the point of contact (Angle = 90°).
• Theorem 2 (Tangents Equality): If PA and PB are tangents from an external point P to a circle with points of contact A and B, then PA = PB.
• Supplementary Angles Property: The angle between the two tangents drawn from an external point and the angle subtended by the line segments joining the points of contact to the centre are supplementary (their sum is 180°).
• Pythagoras Theorem in Circles: In right triangle OAP (where OA is the radius, AP is the tangent, and OP connects the centre to external point P):
• OP² = OA² + AP²
• AP = √(OP² − OA²)

4. Solved Board-Style Examples

Example 1: Finding the Length of a Tangent

Question: A point P is 13 cm away from the centre of a circle O. The radius of the circle is 5 cm. Find the length of the tangent drawn from point P to the circle.

Solution:

1. Let the tangent from P touch the circle at point T.
2. By Theorem 10.1, the radius OT is perpendicular to tangent PT at the point of contact T.

Therefore, ∠OTP = 90°.

1. Triangle OTP is a right-angled triangle where OP is the hypotenuse.

• OP = 13 cm
• OT = 5 cm (radius)

1. Using Pythagoras Theorem:

OP² = OT² + PT²

13² = 5² + PT²

169 = 25 + PT²

PT² = 169 − 25 = 144

PT = √144 = 12 cm

Answer: The length of the tangent is 12 cm.

Example 2: Supplementary Angles Property

Question: Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

Solution:

1. Let P be an external point, and let PA and PB be two tangents touching the circle at A and B. Let O be the centre.
2. Join OA and OB.
3. By Theorem 10.1:

• OA ⊥ PA ⇒ ∠OAP = 90°
• OB ⊥ PB ⇒ ∠OBP = 90°

1. Consider the quadrilateral OAPB. The sum of all four angles in any quadrilateral is 360°:

∠APB + ∠OAP + ∠AOB + ∠OBP = 360°

∠APB + 90° + ∠AOB + 90° = 360°

∠APB + ∠AOB + 180° = 360°

∠APB + ∠AOB = 360° − 180° = 180°

1. Since their sum is 180°, the angles are supplementary.

(Proved)

Example 3: Quadrilateral Circumscribing a Circle

Question: A quadrilateral ABCD is drawn to circumscribe a circle. Prove that:

AB + CD = AD + BC

Solution:

1. Let the circle touch the sides AB, BC, CD, and DA at points P, Q, R, and S respectively.
2. By Theorem 10.2, the lengths of tangents drawn from an external point are equal:

• From point A: AP = AS --- (Equation 1)
• From point B: BP = BQ --- (Equation 2)
• From point C: CR = CQ --- (Equation 3)
• From point D: DR = DS --- (Equation 4)

1. Add all four equations:

(AP + BP) + (CR + DR) = (AS + BQ + CQ + DS)

1. Regroup the terms on both sides:

• (AP + BP) = AB
• (CR + DR) = CD
• (AS + DS) = AD
• (BQ + CQ) = BC

1. Substituting these gives:

AB + CD = AD + BC

(Proved)

Example 4: Parallelogram Circumscribing a Circle is a Rhombus

Question: Prove that a parallelogram circumscribing a circle is a rhombus.

Solution:

1. Let ABCD be a parallelogram circumscribing a circle.
2. Since ABCD is a parallelogram, opposite sides are equal:

• AB = CD --- (Equation 1)
• AD = BC --- (Equation 2)

1. From the property of a circumscribing quadrilateral (as proved in Example 3):

AB + CD = AD + BC

1. Substitute CD with AB, and BC with AD using Equations 1 and 2:

AB + AB = AD + AD

2 × AB = 2 × AD

AB = AD

1. Since adjacent sides are equal (AB = AD) and opposite sides are equal, all four sides must be equal:

AB = BC = CD = DA

1. A parallelogram with all four sides equal is a rhombus.

(Proved)

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid:

• Confusing the Hypotenuse: In right-angled triangle OAP (where P is external, A is the point of contact, O is centre), OP is always the hypotenuse, not the tangent PA. Never write PA² = OP² + OA².
• Misunderstanding Tangent Count: Students often forget that no tangent can be drawn from a point located inside the circle.
• Incorrect Grouping in Proofs: When adding equations for quadrilateral problems (AP = AS, BP = BQ, etc.), make sure the parts forming the same side (like AP and BP) are placed on the same side of the equals sign before adding.

Quick Revision Points:

• 1 point of contact = Tangent.
• 2 points of intersection = Secant.
• Radius is always perpendicular to tangent at the point of contact (∠ = 90°).
• Tangents from an outside point are equal in length (PA = PB).
• Tangents drawn at the ends of a diameter are always parallel.
• Center line OP bisects the angle between the two tangents (∠APO = ∠BPO) and the angle subtended at the center (∠AOP = ∠BOP).