QUIZ TIME : easy question : area of triangle
The plane 6x+4y+2z=12 intersects the coordinate planes to form three sides of a triangle. Find the area of the triangle.
The challenge here is to show as many ways as you can find to get the correct answer. Give exact values if you can, eg. sqrt(14) instead of 3.742.
NOTE to readers: Please award a medal to someone who presents a formula or method that you would not have used.

- mathmate

- jamiebookeater

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- mathmate

##### 1 Attachment

- amistre64

x=0, get the intercepts; put them in a distance formula .... di this for each variable to zero out. then you have the side measures and can do a heron on it

- amistre64

or... divide by 12 and divide off the tops to get the intercepts if we dont have time to do them one by one

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## More answers

- amistre64

6x+4y+2z=12
12 12 12
/6 /4 /2
-------------
2 3 6 are our intercepts

- amistre64

one way is we can take these and form vectors to get an angle to do a sin area formula with

- anonymous

3.164 squre unit approx using area = squrt{ s(s-a)(s-b)(s-c)) ;s=(a+b+c)/2

- amistre64

(2,0,0) (0,0,6)
-(0,3,0) -(0,3,0)
------- -------
<2,-3,0> <0,-3,6>
cos(yaxis) = <2,-3,0>.<0,-3,6> 9
----------------- = ---------------
|<2,-3,0>| |<0,-3,6>| sqrt(4+9+9+36)
cos(y) = 9/sqrt(58)
y = cos-1(9/sqrt(58))
|<2,-3,0>| |<0,-3,6>| sin(cos-1(9/sqrt(58))) / 2
9/sqrt(58) sin(cos-1(9/sqrt(58))) / 2
maybe :)

- mathmate

So far I see Heron's formula from Amistre64 and arijit, and vectors from Amistre64.
Please provide detailed calculations and answers.

- amistre64

my error is in the sqrt(58) :)
sqrt(4+9) * sqrt(9+36)
sqrt(13*45)
3 sqrt(65)

- amistre64

Base * (..... ......height..........) /2
9/3sqrt(65) sin(cos-1(9/3sqrt(65))) / 2
3/sqrt(65) sin(cos-1(3/sqrt(65))) / 2
hopefully lol

- anonymous

PLEASE SEE THE DRAWING........

##### 1 Attachment

- amistre64

i got a few more errors, but the concepts seems solid ;)

- mathmate

So 2,3,6 are the intercepts,
sqrt(13), sqrt(45), sqrt(40) are the sides.

- amistre64

right, and you can heron that for sheer simplicity

- mathmate

Some "exact" answers would be nice! :)

- mathmate

Agree, we also want as many different ways as possible.

- amistre64

|dw:1324567464017:dw|

- amistre64

sqrt(45) sin(y) = height
sqrt(13) = base

- amistre64

b*h/2 = 3sqrt(14) if i did it right that time

- mathmate

Wow! First exact answer!
3sqrt(14) (that's what I have too).
So far we have
intercepts: 2,3,6
sides : sqrt(13), sqrt(45), sqrt(40)
Area : 3sqrt(14)
Methods:
Heron : Amistre64, arigit
Vectors: Amistre64 (to be completed)
bh/2 : Amistre64

- amistre64

the bh/2 is after the vectors give us a sin for an angle to determine the height

- amistre64

i wonder if its worth it to do a 3d integration :) or at least take the measures and translate them to a xy congruent triangle

- phi

\[A= \frac{1}{2}\sqrt{|x|^2|y|^2-(x \cdot y)^2}\]
with (using amistre's two vectors) x=<2,-3,0> y= <0,-3,6>
gives 0.5sqrt( 13*45-81)= 1.5sqrt(65-9)= 1.5sqrt(56)= 3sqrt(14)

- amistre64

work in progress here
|dw:1324568575430:dw|
\[\int_{0}^{6}distance.x.to.y\ dz\]

- amistre64

\[X_i=\left(\frac{z-6}{3},0,z\right)\]
\[Y_i=\left(0,\frac{z-6}{2},z\right)\]
\[X_i-Y_i=\left(\frac{z-6}{3},\frac{-z+6}{2},0\right)\]
distance from X to Y:\[\sqrt{(\frac{z-6}{3})^2+(\frac{-z+6}{2})^2}\]
\[\sqrt{\frac{z^2-6z+36}{9}+\frac{z^2-6z+36}{4}}\]
\[\sqrt{\frac{4z^2-4.6z+4.36+9z^2-9.6z+9.36}{36}}\]
\[\frac{\sqrt{13z^2-78z+468}}{6}\]
ergo lol
\[\int_{0}^{6} \frac{\sqrt{13z^2-78z+468}}{6}dz\]

- amistre64

hmm, wolf says that about 19, which is a bit off from the 3sqrt(14)
i wonder why

- amistre64

348 not 468 ...
but still thats off

- amistre64

oh well, it was a thought. guess i dont know how to freehand integrals yet :)

- mathmate

OK, here's what we have so far (let me know if I missed anything)
Phi: does the formula come from cross product, it's neat, like a generalized form of bh/2.
Related to that, I suggest the cross product of Amistre64's two vectors.
(1/2)AxB=(1/2)<0,-3,6>x<2,-3,0>=(1/2)sqrt(504)=3sqrt(14)
Methods:
1. Heron : Amistre64, arigit
2. (1/2)xysin(theta) + bh/2 : Amistre64
3. integration : Amistre64 (in progress)
4. (1/2)sqrt(|x|^2|y|^2-x.y) : phi
5. half magnitude of cross-product, (1/2)|PxQ|: mathmate

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