F.5 Maths - Ans.fyi 參考答案

F.5 Maths

2015-02-10 8:21 am

Picture: http://postimg.org/image/mixflp3fj/

In the figure, PQR is a right-angled triangle, where angle R=90° and PR=5a cm.
RR1(RR1 means the length RR1 in the picture) and R1R2 are the perpendiculars drawn from R to PQ and from R1 to QR respectively, where PR1= 3a cm.
a) express the lengths of RR1 and R1R2 in terms of a.
b) the perpendiculars R2R3, R3R4, R4R5, ... are drawn in the same way as RR1
and R1R2. Express the length of Rn R(n+1) in terms of a and n.

I did (a) by using similar triangles but the method is complicated, so I cannot find
the pattern of the length Rn R(n+1). Is there any easier methods to do (a) so that
can do (b) simply?

Please help. Thank you very much!!

回答 (1)

邊位都好

2015-02-10 7:00 pm

✔ 最佳答案

Let ∠PRR₁ be θ, therefore, ∠RR₁R₂＝∠R₁R₂R₃＝∠R₂R₃R₄＝⋯⋯＝∠Rn-1RnRn+1＝θand, sin θ＝PR₁/PR＝3/5＝0.6, therefore, cos θ＝4/5＝0.8As RR₁＝PR cos θ＝0.8 PRR₁R₂＝RR₁ cos θ＝0.8 RR₁＝0.8² PRR₂R₃＝R₁R₂ cos θ＝0.8 R₁R₂＝0.8³ PRR₃R₄＝R₂R₃ cos θ＝0.8 R₂R₃＝0.8⁴ PR⋯⋯RnRn+1＝0.8ⁿ⁺¹ PR
(a) RR₁＝0.8 PR＝4aR₁R₂＝0.8² PR＝3.2a
(b) RnRn+1＝0.8ⁿ⁺¹ PR＝5a*0.8ⁿ⁺¹ (or 4a*0.8ⁿ)
(c) Yes, it is a G.S. with common ratio 0.8, first term RR₁ is 3.2a.

2015-02-10 19:07:06 補充：
Sorry, the last line is wrongly written, it should be
first term R₁R₂ is 3.2a.

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