HSC 2014 MX2 Marathon (archive) (1 Viewer)

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dunjaaa

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Re: HSC 2014 4U Marathon

On the topic of triangles, here's a nice question I made: Screen shot 2014-09-30 at 12.31.43 PM.png
 
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integral95

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Re: HSC 2014 4U Marathon

On the topic of triangles, heres a nice question I made: View attachment 30967
Well clearly r is at least positive since you can't have negative sides :p (am I missing something else?)

r>1 ??

for part ii

let the sides be a,ar,ar^2

since
the triangle is right-angled then the Pythagoras theorem applies

(assuming r >1, implying ar^2 is the longest side)

a^2r^4= a^2r^2a^2

and then take the positive square root.
 
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Re: HSC 2014 4U Marathon

Well clearly r is at least positive since you can't have negative sides :p (am I missing something else?)

r>1 ??

for part ii

let the sides be a,ar,ar^2

since
the triangle is right-angled then the Pythagoras theorem applies

(assuming r >1, implying ar^2 is the longest side)

a^2r^4= a^2r^2a^2

and then take the positive square root.
Why can't r be a fraction?
 
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Re: HSC 2014 4U Marathon

well it can haha

I just realised there are 2 sets of solutions

Another case is where a^2 is the longest side instead (where 0<r<1)


And you'd be solving
Exactly, you get 2 r values. However question says value of r not values, so I am assuming we have done something wrong in the first part, unless dunjaa has missed out a s in the word value
 

dunjaaa

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Re: HSC 2014 4U Marathon

You got the second part of the question right, but the first part is wrong. Clearly, r>0, but the sides must also obey a condition.
 

integral95

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Re: HSC 2014 4U Marathon

Exactly, you get 2 r values. However question says value of r not values, so I am assuming we have done something wrong in the first part, unless dunjaa has missed out a s in the word value
value(s) heh
 

integral95

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Re: HSC 2014 4U Marathon

You got the second part of the question right, but the first part is wrong. Clearly, r>0, but the sides must also obey a condition.
Oh great it's not the triangular inequality is it.....

so like
 
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dunjaaa

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Re: HSC 2014 4U Marathon

You got the upper bound correct, but the lower bound is not 0
 

dunjaaa

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Re: HSC 2014 4U Marathon

That's correct, good job :). It's a cool result since it ties in with the golden ratio (Kepler Triangle)
 
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Re: HSC 2014 4U Marathon

Do you just sub in x=b and x=c, show that P(b) and P(c) equals zero, meaning that (x-b) and (x-c) are both factors, implying that their product, the given quadratic, is a factor?
Yeap exactly what i did :p

I just wanted to see if anyone was going to pick up on that, because the polynomial looks "scary" however easy question.
 
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