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HSC 2013-14 MX1 Marathon (archive) (3 Viewers)

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Sy123

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Sy123

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Re: HSC 2013 3U Marathon Thread

Great question I came across:



 
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Re: HSC 2013 3U Marathon Thread

Sy, that is a very VERY nice question. I will post solution tomorrow if no one has it.
 

nightweaver066

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Re: HSC 2013 3U Marathon Thread

I wouldn't say it's great... unless i'm doing something wrong.
 
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Whoever made this question up must have tried so many numbers...or they're just geniuses. Or this is related to some higher mathematical idea.
 
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Sy123

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Whoever made this question up must have tried so many numbers...or they're just geniuses. Or this is related to some higher mathematical idea.
What I did was find cosalpha cosbeta and cosgamma (in terms of numbers)
Then show that 2 of them satisfy cos2theta =2cos^2 theta -1

======

As for how they made that question, if we choose to find the ratio of sides in a triangle where 1 angle is twice the other
We just use the relation:



Then after some manipulation



We just pick a 'reference number' such as a=1



Then we find a pair of b and c such that they match the equation, so b=6/4 c=5/4

Therefore the sides are in ratio 4 : 5 : 6 since similar triangles make it so that the actual numbers doesn't matter as long as we keep the same angles.
Also I'm sure they had to do some constructions by hand and notice that similarity
 

Sy123

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nightweaver066

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Re: HSC 2013 3U Marathon Thread

Great question I came across:



Since we know the sides are in that ratio, we know that all triangles following that are similar so they all have the same corresponding angles.

Lets use the cos rule to determine the smallest angle (angle opposite smallest side) and to determine the largest angle (angle opposite largest side).

You find the largest angle is twice the smallest angle.

Don't think you guys would like this answer :p
 
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Re: HSC 2013 3U Marathon Thread

Since we know the sides are in that ratio, we know that all triangles following that are similar so they all have the same corresponding angles.

Lets use the cos rule to determine the smallest angle (angle opposite smallest side) and to determine the largest angle (angle opposite largest side).

You find the largest angle is twice the smallest angle.

Don't think you guys would like this answer :p
That's the boring short approach :p
 
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Re: HSC 2013 3U Marathon Thread

Most definitely haha.
 

Sy123

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Re: HSC 2013 3U Marathon Thread

Just nit-picking - should there be a restriction on n?
Naturally any n-sided polygon has 3 or more sides otherwise it wouldn't be a polygon :p
 
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