(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...

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°