(a) Simplify : (frac{frac{1}{3}c^{2} – frac{2}{3}cd}{frac{1}{2}d^{2} – frac{1}{4}cd})
(b)
In the diagram, YPF is a straight line. < XPY = 44°, < MPF = 46°, < XYP = < MFP = 90°, /XY/ = 7cm and /MP/ = 9 cm.
(i) Calculate, correct to 3 significant figures, /XM/ and /YF/ ; (ii) Find < XMP.
Explanation
(a) (frac{frac{1}{3}c^{2} – frac{2}{3}cd}{frac{1}{2}d^{2} – frac{1}{4}cd})
= (frac{frac{1}{3}c(c – 2d)}{frac{1}{4}d(2d – c)})
= (frac{frac{1}{3}c (c – 2d)}{-frac{1}{4}d (c – 2d)})
= (-frac{4c}{3d})
(b)(i) In (Delta XYP),
(frac{7}{XP} = sin 44°)
(XP = frac{7}{sin 44})
= (10.08 cm)
< XPM + 44° + 46° = 180°
< XPM = 90°.
In (Delta XPM),
(XM^{2} = XP^{2} + PM^{2})
(XM^{2} = (10.08)^{2} + 9^{2})
= (101.6064 + 81 = 182.6064)
(XM = sqrt{182.6064})
= (15.513 cm approxeq 13.5 cm)
In (Delta XYP),
(frac{7}{YP} = tan 44)
(YP = frac{7}{tan 44})
(YP = 7.249 cm approxeq 7.2 cm)
In (Delta MPF),
(frac{PF}{9} = cos 46)
(PF = 9 cos 46 = 6.2519 cm)
(YF = YP + PF = 7.249 + 6.2519)
= (13.5009 cm approxeq 13.5 cm)
(ii)
(frac{10.1}{9} = tan theta)
(tan theta = 1.122)
(theta = tan^{-1} (1.122))
= 48.296°