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Circles (Tangents, Chords, Arcs)

Introduction

Circles अक्सर aptitude tests और geometry problems में दिखते हैं। radius, diameter, tangent, chord, arc और sector की properties समझने से आप area, length और angle वाले questions जल्दी solve कर सकते हैं।

यह pattern उन relationships पर focus करता है जैसे radius ⟂ tangent at point of contact, किसी external point से निकली equal tangents, और arcs व sectors से जुड़े calculations।

Pattern: Circles (Tangents, Chords, Arcs)

Pattern

Key concept: Circle की basic properties (radius, diameter, tangent perpendicular to radius, equal tangents, arc-sector relations) का उपयोग करके geometry statements को algebraic relations में बदलें और lengths/areas/angles निकालें।

Important facts:
• Radius हमेशा tangent पर point of contact पर ⟂ होता है।
• किसी external point से निकली tangents बराबर होती हैं।
• Central angle (θ) जिस arc को subtend करता है उसकी length = (θ/360) × 2πr और sector area = (θ/360) × πr² होती है।
• Chord center पर equal angles subtend करता है; center से chord पर डाला perpendicular chord को bisect करता है।
• Diameter = 2 × radius; semicircle की properties कई angle problems को आसान करती हैं।

Step-by-Step Example

Question

एक circle का radius 14 cm है। उसके एक sector का central angle 60° है। निकालें: (a) arc की length, और (b) sector का area। (Use π = 22/7)

Solution

  1. Step 1: Given values और formulas पहचानें।

    Radius r = 14 cm, Central angle θ = 60°.
    Arc length formula: Arc = (θ/360) × 2πr.
    Sector area formula: Area = (θ/360) × πr².

  2. Step 2: Arc length निकालें।

    Arc = (60/360) × 2 × (22/7) × 14.
    Stepwise simplify करें: 60/360 = 1/6, 2 × (22/7) × 14 = 2 × 22 × 2 = 88.
    इसलिए Arc = (1/6) × 88 = 88/6 = 44/3 ≈ 14.67 cm.

  3. Step 3: Sector area निकालें।

    Area = (60/360) × (22/7) × 14².
    60/360 = 1/6; 14² = 196. तो Area = (1/6) × (22/7) × 196.
    Simplify: (22/7) × 196 = 22 × 28 = 616. फिर Area = (1/6) × 616 = 616/6 = 308/3 ≈ 102.67 cm².

  4. Final Answer:

    (a) Arc length = 44/3 cm (≈ 14.67 cm).
    (b) Sector area = 308/3 cm² (≈ 102.67 cm²).

  5. Quick Check:

    • Angle fraction = 60/360 = 1/6 - arc और area दोनों full circle values के 1/6 हैं।
    • Full circumference = 2πr = 88 → 1/6 of 88 = 14.67 (same as arc).
    • Full circle area = πr² = 616 → 1/6 of 616 = 102.67 (same as sector area) ✅

Quick Variations

1. Arc length दी हो तो central angle या radius निकालें।

2. Tangent-radius perpendicular वाले questions: right-angle relations से lengths निकालें।

3. External point से निकली equal tangents: दोनों lengths equal रखकर equations बनाएं।

4. दो concentric circles के बीच वाले ring (annulus) का area: π(R² - r²)।

Trick to Always Use

  • Step 1 → Angle को जल्दी से fraction (θ/360) में बदलें ताकि arc और area दोनों में काम आए।
  • Step 2 → Tangents हों तो equal lengths mark करें और subtraction से unknown निकालें।
  • Step 3 → Center से chord पर perpendicular हमेशा chord को bisect करता है - right triangle से half-chord length निकालें।

Summary

Summary

Circles (Tangents, Chords, Arcs) pattern में:

  • θ/360 arc length और sector area का key fraction है।
  • Radius हमेशा tangent पर perpendicular होता है - इसी से right triangles मिलते हैं।
  • External point से निकली equal tangents कई length problems को आसान बनाती हैं।
  • Quick checks: sector/arc formulas में full circumference / full area का fraction compare करें।

Practice

(1/5)
1. A circle has a radius of 7 cm. Find its circumference. (Use π = 22/7)
easy
A. 44 cm
B. 49 cm
C. 42 cm
D. 40 cm

Solution

  1. Step 1: Recall the formula.

    Circumference = 2πr.

  2. Step 2: Substitute values.

    C = 2 × (22/7) × 7.

  3. Step 3: Simplify and compute.

    7 cancels with 7, so C = 2 × 22 = 44 cm.

  4. Final Answer:

    Circumference = 44 cm → Option A.

  5. Quick Check:

    Full circumference 2πr with r=7 gives 88/2 = 44 for π=22/7 ✅
Hint: Multiply radius by 2π to get circumference.
Common Mistakes: Using πr² instead of 2πr.
2. The radius of a circle is 10 cm. Find the area of the circle. (Use π = 3.14)
easy
A. 300 cm²
B. 314 cm²
C. 320 cm²
D. 3140 cm²

Solution

  1. Step 1: Recall the formula.

    Area = πr².

  2. Step 2: Substitute values.

    Area = 3.14 × 10 × 10.

  3. Step 3: Simplify and compute.

    Area = 3.14 × 100 = 314 cm².

  4. Final Answer:

    Area = 314 cm² → Option B.

  5. Quick Check:

    Square radius first (100) then multiply by π = 3.14 → 314 ✅
Hint: Square the radius and multiply by π.
Common Mistakes: Using 2πr instead of πr² for area.
3. A tangent is drawn to a circle of radius 5 cm from an external point. If the distance between the point and the center is 13 cm, find the length of the tangent.
easy
A. 12 cm
B. 10 cm
C. 8 cm
D. 9 cm

Solution

  1. Step 1: Use the right-triangle relation.

    The radius to the point of contact is perpendicular to the tangent. So OP² = r² + PT² where OP = distance from center to external point and PT = tangent length.

  2. Step 2: Substitute values.

    PT² = OP² - r² = 13² - 5² = 169 - 25.

  3. Step 3: Compute and take square root.

    PT² = 144 ⇒ PT = √144 = 12 cm.

  4. Final Answer:

    Length of tangent = 12 cm → Option A.

  5. Quick Check:

    5² + 12² = 25 + 144 = 169 = 13² ✅
Hint: Use PT = √(OP² - r²) where OP is center-to-point distance.
Common Mistakes: Adding r² instead of subtracting it when isolating PT².
4. Find the length of the arc subtending an angle of 90° at the center of a circle with radius 14 cm. (Use π = 22/7)
medium
A. 28 cm
B. 33 cm
C. 22 cm
D. 44 cm

Solution

  1. Step 1: Recall the arc length formula.

    Arc length = (θ/360) × 2πr.

  2. Step 2: Substitute given values.

    θ = 90°, r = 14 ⇒ Arc = (90/360) × 2 × (22/7) × 14.

  3. Step 3: Simplify and compute.

    90/360 = 1/4; 2 × (22/7) × 14 = 88; (1/4) × 88 = 22 cm.

  4. Final Answer:

    Arc length = 22 cm → Option C.

  5. Quick Check:

    One-fourth of full circumference (88) is 22 ✅
Hint: Arc = (θ/360) × circumference.
Common Mistakes: Using πr² (area) instead of arc length formula.
5. The length of a tangent drawn from an external point to a circle is 12 cm. If the radius of the circle is 9 cm, find the distance between the external point and the center of the circle.
medium
A. 12 cm
B. 18 cm
C. 20 cm
D. 15 cm

Solution

  1. Step 1: Understand the geometry.

    The radius to the point of contact is perpendicular to the tangent. This forms a right triangle with radius (r), tangent (PT), and line from the center to external point (OP).

  2. Step 2: Apply Pythagoras theorem.

    OP² = r² + PT².

  3. Step 3: Substitute given values.

    OP² = 9² + 12² = 81 + 144 = 225.

  4. Step 4: Take square root.

    OP = √225 = 15 cm.

  5. Final Answer:

    Distance between external point and center = 15 cm → Option D.

  6. Quick Check:

    9² + 12² = 15² → 81 + 144 = 225 ✅
Hint: Use Pythagoras theorem: OP = √(r² + tangent²).
Common Mistakes: Subtracting instead of adding under the square root.

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