"making up" geo theorems (1 Viewer)

[Giant Lobster]

Do i have to prove anything I use thats outside the standard set of theorems im given?

e.g. ill give a specific example: I know this is true, that if 2 angle bisectors meet at a point, and that point is joined to the third vertex, that angle is bisected (concurrency of angle bisectors) but proving this in an exam will waste so much space / time.

Also, im not sure if this theorem is usable directly; the point of intersection of the angle bisectors (is this called the centroid? i dont know the geo technicalities very well) of a triangle divides the vertex angle bisectors from vertex to opposite side in ratio of 2:1.

Such theorems take really long (for me anyway) to prove and im not sure if im even capable of doing so in an exam situation but I need to use them quite often.

My teachers are slack, always takes marks off cos i make up theorems

And also, circ geo is a bitch. Is there a systematic way of doing it? my current failure rate at 4u geo is like 50%

 

[spice girl]

the pt of intersection of angle bisectors is called the 'incentre', it's the centre of the 'incircle', or the circle that has the 3 sides of the triangle as tangents. If you inscribe a circle in a triangle (i.e. put a circle inside a triangle), and connect the centre of circle to the vertices, u can see why these lines bisect the angles: congruent triangles everywhere!

centroid is the intersection of the medians (median is line joining vertex with midpt of opposite side). the 2:1 ratio can somehow be proved using vectors (never remembered how)

and as for making up theorems, dun do it, unless you know wot u're talking about.

 

[Grey Council]

hrm, i know you can't do that in 3u maths, dunno about 4u maths.

EDIT:
By that i mean:

Do i have to prove anything I use thats outside the standard set of theorems im given?

in 3u maths you hafta prove anything outside the syllabus/ anything beyond the basic theorems. Maybe it's different in 4u, but i doubt it.

 

[Giant Lobster]

ahh yes the incentre, i totaly forgot about that.
well if i do need to prove everything i use, how do other ppl do it? every circ geo qu wud then take so very long... im thinkin i cud do qu 1 2 and possibly 3 before i do a qu 7 - 8 circ geo question... or is that a tad exaggerated.. no, its happened before.

 

[turtle_2468]

abbreviation.
Such as Vert.opp.(angle sign) for reasons, and supp. (angle sign) instead of "opposite angles of a cyclic quadrilateral are supplementary"

 

[Giant Lobster]

Yes! abbrev.
however my gay school threatened to take marks off next time cos i abbrev angle into > in my ext1 test
I was fully begging for mercy to not take marks off cos of abbrev... after much philosophizing about what maths is really about and the stupidity of not being able to abbrev, they let me off.

To the point, my school doesnt allow me to abbrev. and ive been told by my teacher that in the hsc they wont let you either. Not sure if theyre right on that tho, i dont see anything wrong with correct abbreviating.

 

[Constip8edSkunk]

like seriously..... maybe they should go teach hsc sciences instead...

 

[Giant Lobster]

well its no matter cos all the other students have to deal with it too so i guess its fair
but i just hope im allowed to abbreviate everything in the real thing, saving precious seconds which contribute to heavy pondering when q8 comes

edit:: whilst on the topic of abbreviating, my teachers go we cant use cis@ in the hsc; is that true? (of course I would investigate this further so dont feel guilty for not being sure ) And apparently in induction step 4 we have to write the full thing in even more detail. i.e. "Since its true for n = 1 then it's true for n = 1 + 1 = 2 and thus true for n = 2 + 1 = 3. Hence..." (Yes, we need the 1 + 1 according to my teacher) lawd...
Can someone just type a typical step 4 for me, i wanna see the detail required. thanx

 

[KeypadSDM]

Originally posted by Giant Lobster
edit:: whilst on the topic of abbreviating, my teachers go we cant use cis@ in the hsc; is that true? (of course I would investigate this further so dont feel guilty for not being sure ) And apparently in induction step 4 we have to write the full thing in even more detail. i.e. "Since its true for n = 1 then it's true for n = 1 + 1 = 2 and thus true for n = 2 + 1 = 3. Hence..." (Yes, we need the 1 + 1 according to my teacher) lawd...
Can someone just type a typical step 4 for me, i wanna see the detail required. thanx

Cis(Theta) = e ^ [i (Theta)]

The latter is ALWAYS acceptable.

Therefore true for n = k + 1 if true for n = k
But true for n = 1
:. True for n = 2
:. True for n = 3
:. True for all n >= 1 (n E Z)

The last line could also be:

:. True for all integral n >= 1

 

[Giant Lobster]

ahh right thanx

And yes I know of that exp(i@) = cis@, thats one of eulean's identities. But its not in the syllabus, so to replace everything cis with e^ must I prove it first?

 

[Constip8edSkunk]

hahaha the math department at my school had a huge argument about would is acceptable for the last line for the hsc. for the sake of the hsc and to avoid the wrath of the markers just add say, s(n) for n=1 hence by principle of induction, it is true for all n.

apparently the cis@ notation isnt even in the hsc syllabus, but its accepted in the hsc as all the text books uses it. as for the e^i@, naming it the first time you use it should b enough. then its best to avoid to many out of the syllabus stuff ... esp with simple questions as theres always risks. the bos was particularly vague about the markings of such usages. (though it really shouldnt be much of an issue)

 

[KeypadSDM]

Originally posted by Giant Lobster
And yes I know of that exp(i@) = cis@, thats one of eulean's identities. But its not in the syllabus, so to replace everything cis with e^ must I prove it first?

If you can prove it, you're miles ahead of me.

I used it in the paper this year, and it payed off timewise, and I didn't prove it, nor state it. I just used it.

 

[Grey Council]

lol, there you go. And in case you are living in a dream world, KeypadSDM came first in thestate this year (i think teeheheehe).

 

[spice girl]

let z = f(@) = cos@ + isin@
then f'(@) = -sin@ + icos@

the integral I{ f'(@)/f(@) }d@ = ln[f(@)]
notice that f'(@)/f(@) = i
thus I{ i }d@ = ln(z)
i@ = ln(z)
z = e^i@

no point using e^i@ instead of cis@ tho...

 

[turtle_2468]

all uni ppl expect you to use e^i@ tho..

 

[Giant Lobster]

Originally posted by spice girl
let z = f(@) = cos@ + isin@
then f'(@) = -sin@ + icos@

the integral I{ f'(@)/f(@) }d@ = ln[f(@)]
notice that f'(@)/f(@) = i
thus I{ i }d@ = ln(z)
i@ = ln(z)
z = e^i@

no point using e^i@ instead of cis@ tho...

yeh i love the proof for this identity... so elegantly simple
Im pondering now, if they ask me to prove de moivre's, without any mention of messy induction I would surely prove this identity first, then raise cis to a power and show its true like that.

actually using this u can prove de moivres works for all numbers, not just integers, which is why induction is a dodgy proof

 

[KeypadSDM]

Originally posted by spice girl
thus I{ i }d@ = ln(z)
i@ = ln(z)
z = e^i@

Just being technical:

{ i }d@ = ln(z)
i@ + c = ln(z)
z = e^(i@ + c)

Note: @ = 0, z = 1

:. 1 = e^c
:. c = 0

:. z = e^(i@)