Find ${\rm{A\hat BC}}$.

Find the exact area of the sector BDC.

METHOD 1

correct substitution into formula for area of triangle     (A1)

eg$\,\,\,\,\,$$\frac{1}{2}(6)\left( {2\sqrt 3 } \right)\sin B,{\text{ }}6\sqrt 3 \sin B,{\text{ }}\frac{1}{2}(6)\left( {2\sqrt 3 } \right)\sin B = 3\sqrt 3$

correct working     (A1)

eg$\,\,\,\,\,$$6\sqrt 3 \sin B = 3\sqrt 3 ,{\text{ }}\sin B = \frac{{3\sqrt 3 }}{{\frac{1}{2}(6)2\sqrt 3 }}$

$\sin B = \frac{1}{2}$    (A1)

$\frac{\pi }{6}(30^\circ )$    (A1)

${\rm{A\hat BC}} = \frac{{5\pi }}{6}(150^\circ )$     A1     N3

METHOD 2

(using height of triangle ABC by drawing perpendicular segment from C to AD)

correct substitution into formula for area of triangle     (A1)

eg$\,\,\,\,\,$$\frac{1}{2}\left( {2\sqrt 3 } \right)(h) = 3\sqrt 3 ,{\text{ }}h\sqrt 3$

correct working     (A1)

eg$\,\,\,\,\,$$h\sqrt 3  = 3\sqrt 3$

height of triangle is 3     A1

${\rm{C\hat BD}} = \frac{\pi }{6}(30^\circ )$    (A1)

${\rm{A\hat BC}} = \frac{{5\pi }}{6}(150^\circ )$     A1     N3

[5 marks]

recognizing supplementary angle     (M1)

eg$\,\,\,\,\,$${\rm{C\hat BD}} = \frac{\pi }{6},{\text{ sector}} = \frac{1}{2}(180 - {\rm{A\hat BC)(}}{{\text{6}}^2})$

correct substitution into formula for area of sector     (A1)

eg$\,\,\,\,\,$$\frac{1}{2} \times \frac{\pi }{6} \times {6^2},{\text{ }}\pi ({6^2})\left( {\frac{{30}}{{360}}} \right)$

${\text{area}} = 3\pi {\text{ }}({\text{c}}{{\text{m}}^2})$     A1     N2

[3 marks]

In part (a) of this question, the large majority of candidates substituted correctly into the area formula for the triangle, though algebraic errors kept some of them from simplifying the equation to $\sin {\rm{A\hat BC = }}\frac{1}{2}$. Unfortunately, a number of candidates who got to this point often did not know the correct angles that correspond with this sine value.

In part (b), many candidates realized that ${\rm{C\hat BD}}$ was the supplement of ${\rm{A\hat BC}}$. However, at this point many candidates substituted 30°, or their follow-through angle in degrees, into the formula for the area of a sector found in the formula booklet, not understanding that this formula only works for angles in radians.