3.69: Triangle with surds: side, sine and height

0606/22/O/N/21 — Question 6 · 7 marks

Do not use a calculator in this question. All lengths are in centimetres.

3.69 diagram
(a)
The diagram shows triangle ABCABC with AC=6−2AC = \sqrt{6}-\sqrt{2}, AB=6+2AB = \sqrt{6}+\sqrt{2} and angle CAB=60∘CAB = 60^{\circ}.
Find the exact length of BCBC.
[3]
(b)
Show that sin ACB=6+24\mathrm{sin}\,ACB = \dfrac{\sqrt{6}+\sqrt{2}}{4}.
[2]
(c)
Show that the perpendicular distance from AA to the line BCBC is 11.
[2]