Hint
Use the minor segment and its stated angle. Use the sector angle \(\displaystyle \frac{\pi }{2}\).
Worked solution
(a)
Subtract triangle area \(\displaystyle \frac{121}{2}\) from sector area \(\displaystyle \frac{121 \pi}{4}\).
Answer: \(\displaystyle - \frac{121}{2} + \frac{121 \pi}{4}\) \(\displaystyle \mathrm{cm}^{2}\)
(b)
The radii form a right-angle triangle for the chord; use \(\displaystyle r\theta\) for the arc.
Answer: \(11 \sqrt{2}\) cm; \(\displaystyle \frac{11 \pi}{2}\) cm
(c)
Use Pythagoras on the two perpendicular radii.
\(242\)
Answer: \(242\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Subtract the triangle area from the matching sector area.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A segment and a sector are different regions.
(b) · 2 marks
- 1 mark: Use the right triangle for the chord and \(\displaystyle r\theta\) for the curved arc.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A chord length is not an arc length.
(c) · 2 marks
- 1 mark: Use Pythagoras on the two perpendicular radii.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A segment and a sector are different regions.
Total: 6 marks.