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Exercise 9.4: Chord and Arcs of a Circle

Download free PDF notes covering radian conversions for arc length ($\theta$ in radians: $\theta = \text{degrees} \times \frac{\pi}{180^\circ}$), arc length formula $l = r\theta$, minor arc calculation (e.g., $r=7$ cm, $\theta=100^\circ$ → $\theta = \frac{5\pi}{9}$ rad → $l_{\text{minor}} = \frac{35\pi}{9} \approx 12.22$ cm), major arc calculation (subtract from circumference $C = 2\pi r$ → $l_{\text{major}} = C - l_{\text{minor}} = \frac{91\pi}{9} \approx 31.76$ cm), semicircle area $A = \frac{1}{2}\pi r^2$ (e.g., $r=2.8$ ft → $A = 3.92\pi \approx 12.32$ ft²), semicircle perimeter $P = \pi r + 2r$ (curved arc + diameter), and composite geometric perimeters - strictly according to FBISE 2026 SLOs.

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Chapter Overview & SLOs

How do we transition from rigid straight-line chords to dynamic curved perimeters, isolating exact boundary lengths and composite perimeters using radian scaling? The final section of Chapter 9 shifts focus from internal chord intersections to the metric properties of a circle's outer curvature.

Radian Conversions and Arc Length Metrics: The absolute length of a curved boundary arc ($l$) is governed directly by $l = r\theta$, where $\theta$ must be in radians.

Degree to Radian Conversion: $\theta = \text{degrees} \times \frac{\pi}{180^\circ}$

Example: $r = 7$ cm, central angle $100^\circ$:

  • $\theta = 100^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{9}$ radians
  • $l_{\text{minor}} = r\theta = 7 \times \frac{5\pi}{9} = \frac{35\pi}{9} \approx 12.22$ cm

Major Arc Calculation: Subtract minor arc from total circumference.

  • $C = 2\pi r = 14\pi \approx 43.98$ cm
  • $l_{\text{major}} = C - l_{\text{minor}} = 14\pi - \frac{35\pi}{9} = \frac{126\pi - 35\\pi}{9} = \frac{91\pi}{9} \approx 31.76$ cm

Semicircular Window Area and Perimeter: For a semicircular window with $r = 2.8$ feet:

  • Area: $A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi(2.8)^2 = 3.92\pi \approx 12.32$ ft²
  • Perimeter: $P = \pi r + 2r = \pi(2.8) + 2(2.8) = 2.8\pi + 5.6 \approx 14.40$ feet

Important Rule: Semicircle perimeter includes both the curved arc ($\pi r$) AND the straight diameter ($2r$).

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

  • How do we convert central subtended angles from degree measurements into circular radian tracking units?
  • How do we calculate exact linear boundary lengths for minor and major circular arcs using radius scaling properties?
  • How do we verify that the additive summation of partitioned major and minor arc lengths equals the circle's total circumference?
  • How do we compute the complete composite perimeter of a semicircle by combining curved arc spans with straight baseline diameters?

Frequently Asked Questions (FAQ)

1. Are these Class 10 Mathematics notes based on the latest FBISE syllabus for 2026?
Yes, these notes are strictly designed according to the Student Learning Outcomes (SLO) provided by the Federal Board (FBISE) for the 2026 academic year. We regularly update our content to match the latest curriculum changes and exam patterns.

2. Do these Mathematics 9 notes include solved exercise questions and diagrams?
Absolutely. These notes contain comprehensive solutions to all textbook exercise questions, including Multiple Choice Questions (MCQs), Short Questions, and detailed Long Questions. We also include labeled diagrams and key definitions to help you secure maximum marks in your board exams.

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