1.51: Past-paper question 3

4PM1/1/November/2025 — Question 3 · 6 marks

1.51 diagram 1
Figure 1 shows triangle ABC\displaystyle ABC where
AB=12cmBC=15cmAC=xcm∠BAC=60∘AB = 12 \mathrm{cm} \quad BC = 15 \mathrm{cm} \quad AC = x \mathrm{cm} \quad \angle BAC = 60^\circ
(a) Show that x\displaystyle x satisfies the equation
x2−12x−81=0x^2 - 12x - 81 = 0
(3)
(b) Hence find the exact area, in cm2\displaystyle \mathrm{cm}^2, of triangle ABC\displaystyle ABC
Give your answer in the form 9(pq+r)\displaystyle 9(p\sqrt{q} + \sqrt{r}) where p,q\displaystyle p, q and r\displaystyle r are integers.
(3)