Theory question 106
Algebraic Expressions(a) Simplify (1/3 c2 − 2/3 cd)/(1/2 d2 − 1/4 cd)
(b) In the diagram, YPF is a straight line. ∠XPY = 44°, ∠MPF = 46°, ∠XYP = ∠MFP = 90°, |XY| = 7cm and |MP| = 9cm. (i) Calculate, correct to 3 significant figures, |XM| and |YF|. (ii) Find ∠XMP
Show the model answer
Model answer
(a) Numerator = c(c − 2d)/3; denominator = d(2d − c)/4. Result = 4c(c − 2d)/[3d(2d − c)] = −4c/(3d).
(b)(i) XP = 7/sin 44° = 10.08 cm; YP = 7/tan 44° = 7.249 cm; PF = 9 cos 46° = 6.252 cm.
YF = 7.249 + 6.252 = 13.5 cm.
∠XPM = 180 − 44 − 46 = 90°, so XM = √(10.082 + 92) = √182.5 ≈ 13.5 cm.
(ii) tan ∠XMP = XP/MP = 10.08/9 = 1.120, so ∠XMP ≈ 48.2°.
Explanation
Notice ∠XPM is a right angle because the three angles at P make a straight line.