Hint

The chord has length \(\displaystyle 2r\sin (\frac{\theta }{2})\). The segment is the sector with its central triangle removed.

Worked solution

(a)

The arc is \(\displaystyle 8(\frac{\pi }{3})= \frac{8\pi }{3}\mathrm{cm}\). The chord is \(\displaystyle 16\sin (\frac{\pi }{6})= 8\mathrm{cm}\).

Answer: Arc \(\displaystyle \frac{8\pi }{3}\text{ cm}\); chord 8 cm.

(b)

The sector area is \(\displaystyle (\frac{1}{2})(64)(\frac{\pi }{3})= \frac{32\pi }{3}\mathrm{cm}^{2}\). The triangle area is \(\displaystyle (\frac{1}{2})(64)\sin (\frac{\pi }{3})= 16\sqrt{3}\mathrm{cm}^{2}\). Subtracting gives \(\displaystyle \frac{32\pi }{3}- 16\sqrt{3}\approx 5.79751\mathrm{cm}^{2}\), or \(\displaystyle 5.80 \mathrm{cm}^{2}\) to 3 significant figures.

Answer: \(\displaystyle \frac{32\pi }{3}- 16\sqrt{3}\text{ cm}^{2}\), approximately \(5.80\text{ cm}^2\).

(c)

The segment is bounded by the chord and the minor arc. Its perimeter is \(\displaystyle 8+ \frac{8\pi }{3}\mathrm{cm}\); the two radii are internal to this description and are not its boundary.

Answer: \(\displaystyle 8+ \frac{8\pi }{3}\text{ cm}\).

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.

(a) · 2 marks

• 1 mark: Give minor arc length \(\displaystyle \frac{8\pi }{3}\mathrm{cm}\).
• 1 mark: Give chord length 8 cm using a correct triangle method.

Alternative methods: The central triangle is equilateral, so the chord equals the radius; this is accepted.

Common error: Treating the chord and arc as equal lengths.

(b) · 3 marks

• 1 mark: Obtain the correct sector and triangle areas.
• 1 mark: Subtract to give the exact segment area \(\displaystyle \frac{32\pi }{3}- 16\sqrt{3}\).
• 1 mark: Give \(\displaystyle 5.80 \mathrm{cm}^{2}\) to 3 significant figures without premature rounding.

Alternative methods: The direct minor-segment formula gives full credit when the exact expression and rounding are correct.

Common error: Subtracting the triangle from the full disk instead of the minor sector.

(c) · 2 marks

• 1 mark: Use chord plus minor arc.
• 1 mark: Give exact perimeter \(\displaystyle 8+ \frac{8\pi }{3}\mathrm{cm}\).

Alternative methods: A clear boundary trace is accepted.

Common error: Adding two radii as though the region were a sector.

Total: 7 marks.