Quod Erat Demonstrandum


Some F.4 textbook questions

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 4:55 下午
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Students may find the following textbook questions difficult.

Question 1

Refer to the figure below, given that
DE \perp AB and DF \perp AC;
ED = a;
AB : AC = m.


Show that \tan\alpha = \frac{2m - 1}{\sqrt{3}}.


To use the given information AB : AC = m, we express AB and AC in terms of \alpha, \beta and a.

AB = AE + EB = \sqrt{3}a + a\tan\alpha
AC = AF + FC = \sqrt{3}a + a\tan\beta (note: DF = a)


m = \frac{AB}{AC} = \frac{\sqrt{3}a + a\tan\alpha}{\sqrt{3}a + a\tan\beta} = \frac{\sqrt{3} + \tan\alpha}{\sqrt{3} + \tan\beta} – – – – – – (*)

Then, many students stop there and don’t know how to find \tan\alpha. May be, students find that, there are, at least, two different approaches.

One is geometry. \tan\alpha, the direct meaning in the figure is \frac{BE}{ED}, but, it is not that easy to express BE explicitly.

Another one is algebra, that is using trigonometric identities to do some algebraic computation so as to figure out the relation between \tan\alpha and m.

We select the latter.

To find \tan\alpha, we need to get rid of \beta, hence all we need is replacing \beta by something involves \alpha. Here is the crucial key: \alpha + \beta = 60^o\Rightarrow \beta = 60^o - \alpha, and (*) will be

m = \frac{\sqrt{3} + \tan\alpha}{\sqrt{3} + \tan(60^o - \alpha)}

Well, it is not a must that students can obtain the required results, because, algebraic computation is still nightmares to some students.

m(\sqrt{3} + \frac{\sqrt{3} - \tan\alpha}{1 + \sqrt{3}\tan\alpha}) = \sqrt{3} + \tan\alpha
\frac{2m(\sqrt{3} + \tan\alpha)}{1 + \sqrt{3}\tan\alpha} = \sqrt{3} + \tan\alpha

Urm, what’s next? Some students may expand further and come up with monsters. But, once we discover that there are similar non-zero terms (\sqrt{3} + \tan\alpha) on both sides, cancel it and the expression above can be further simplied as

\frac{2m}{1 + \sqrt{3}\tan\alpha} = 1
\therefore \tan\alpha = \frac{2m - 1}{\sqrt{3}}

Question 2

This is actually a past-paper question (1992 paper 2), just highlight one of the parts.


A regular pentagon inscribing in a circle with radius r. Prove that

PA^2 + PB^2 + PC^2 + PD^2 + PE^2 = 10r^2


The previous result showing that

PD^2 = 2r^2 - 2r^2\cos(\theta + \frac{6\pi}{5}).

According to the marking scheme, we have


But, as I’d mentioned in class, it seems that the presentation above is not clear enough. Something is missing in the marking scheme. Let me explain.

Refer to the following figure.


By cosine law, it is easy to show that

(PA')^2 = 2r^2 - 2r^2\cos\phi – – – – – – (**)

Imagine that the point A' is moving along the circumference, like


The formula (**) is still valid.

How about when A' moves to some position such that \phi > \pi? Like


Is (**) still valid? Well, we need to show it out! Just refer to the figure above, by cosine law

= 2r^2 - 2r^2\cos\angle POA'
= 2r^2 - 2r^2\cos(2\pi - \phi)
= 2r^2 - 2r^2\cos\phi

Yes, (**) is still valid for \phi > \pi. Hence, we have


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