F4 MATHS TRIGONMETRY

Andrew

2007-04-16 4:15 am

A flagstaff BC is fixed on the top of a tower AB. At a point P on the horizontal ground , an observer finds that the angles of elevation of the point C on the top of the flagstaff and the point B on the top of the tower are 20° AND 12° respectively. On walking a distance of 16 m towards the tower, the observer reaches a point Q AND FINDS THAT ∠CQB=8°.
(a) SHOW THAT B,C,P,&Q ARE CONCYCLIC.
(B)find the heights of
(I) the tower AB,
(ii) the FLAGSTAFF BC.

回答 (1)

風鈴鈴

2007-04-16 4:49 am

✔ 最佳答案

(a) ∠CPB = ∠CPQ - ∠BPQ = 8° = ∠CQB
∴ B, C, P, Q are concyclic (Converse of ∠s in same sagment)
(b) (i) ∠QBA = ∠CPQ = 20° (Ext. ∠, cyclic quad.)
∠BQA = 180° - 90° - 20° = 70°
tan∠BQA = AB/AQ
AQ = AB/tan∠BQA ......(1)
tan∠BPA = AB/(AQ+16) ......(2)
Put (1) into (2),
tan∠BPA = AB/[(AB/tan∠BQA)+16]
[(AB/tan∠70°)+16]tan∠20° = AB
0.13247AB + 5.82352 = AB
AB = 6.71m (Cor. to 3 sig. fig.)
(ii) AQ = AB/tan∠BQA = 2.44224m
tan∠CQA = (AB+BC) / AQ
2.44224tan78 ° = 6.71 + BC
BC = 4.78m (Cor. to 3 sig. fig.)

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