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Circles (Tangents, Chords, Arcs)

Introduction

Circles appear often in aptitude tests and geometry problems. Mastering properties of radius, diameter, tangent, chord, arc, and sector will help you solve area, length, and angle questions quickly.

This pattern focuses on relationships such as radius ⟂ tangent at point of contact, equal tangents from an external point, and calculations involving arcs and sectors.

Pattern: Circles (Tangents, Chords, Arcs)

Pattern

Key concept: Use basic circle properties (radius, diameter, tangent perpendicular to radius, equal tangents, arc-sector relations) to convert geometry statements into algebraic relations and compute lengths/areas/angles.

Important facts:
• Radius ⟂ tangent at the point of contact.
• Tangents drawn from the same external point are equal in length.
• Central angle (θ) subtends an arc of length = (θ/360) × 2πr and sector area = (θ/360) × πr².
• Chord subtends equal angles at the center; Perpendicular from center to chord bisects the chord.
• Diameter = 2 × radius; semicircle properties often reduce angle problems.

Step-by-Step Example

Question

A circle has radius 14 cm. A sector of the circle has a central angle of 60°. Find (a) the length of the arc, and (b) the area of the sector. (Use π = 22/7)

Solution

  1. Step 1: Identify known values and formulas.

    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: Compute arc length.

    Arc = (60/360) × 2 × (22/7) × 14.
    Simplify stepwise: 60/360 = 1/6, 2 × (22/7) × 14 = 2 × 22 × 2 = 88.
    So Arc = (1/6) × 88 = 88/6 = 44/3 ≈ 14.67 cm.

  3. Step 3: Compute sector area.

    Area = (60/360) × (22/7) × 14².
    60/360 = 1/6; 14² = 196. So Area = (1/6) × (22/7) × 196.
    Simplify: (22/7) × 196 = 22 × 28 = 616. Then 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 used = 60/360 = 1/6 - both arc and area are 1/6 of full circumference and full area respectively.
    • Full circumference = 2πr = 88 → 1/6 of 88 = 14.67 (matches arc).
    • Full circle area = πr² = 616 → 1/6 of 616 = 102.67 (matches sector area) ✅

Quick Variations

1. Given arc length, find central angle or radius.

2. Tangent-radius perpendicular problems: use right-angle relations to find lengths.

3. Two equal tangents from an external point: set tangent lengths equal to form equations.

4. Area of ring (annulus) between two concentric circles: π(R² - r²).

Trick to Always Use

  • Step 1 → Convert angles to fraction of 360° early (θ/360) to reuse for arc length and sector area.
  • Step 2 → If tangents from a point are involved, mark equal lengths and use subtraction to find unknowns.
  • Step 3 → For perpendicular from center to chord, remember it bisects the chord - use right triangle relations to find half-chord length.

Summary

Summary

For the Circles (Tangents, Chords, Arcs) pattern:

  • Use θ/360 as the key fraction for arc length and sector area calculations.
  • Radius is perpendicular to tangent at point of contact - use this to form right triangles.
  • Equal tangents from an external point simplify many length problems.
  • Quick checks: compare fraction of full circumference / full area when using sector/arc formulas.

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.

Mock Test

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