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mathmate
 4 years ago
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
 4 years ago
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.

This Question is Closed

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1x=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
 4 years ago
Best ResponseYou've already chosen the best response.1or... divide by 12 and divide off the tops to get the intercepts if we dont have time to do them one by one

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.16x+4y+2z=12 12 12 12 /6 /4 /2  2 3 6 are our intercepts

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1one way is we can take these and form vectors to get an angle to do a sin area formula with

arijit.mech
 4 years ago
Best ResponseYou've already chosen the best response.03.164 squre unit approx using area = squrt{ s(sa)(sb)(sc)) ;s=(a+b+c)/2

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1(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 = cos1(9/sqrt(58)) <2,3,0> <0,3,6> sin(cos1(9/sqrt(58))) / 2 9/sqrt(58) sin(cos1(9/sqrt(58))) / 2 maybe :)

mathmate
 4 years ago
Best ResponseYou've already chosen the best response.1So far I see Heron's formula from Amistre64 and arijit, and vectors from Amistre64. Please provide detailed calculations and answers.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1my error is in the sqrt(58) :) sqrt(4+9) * sqrt(9+36) sqrt(13*45) 3 sqrt(65)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1Base * (..... ......height..........) /2 9/3sqrt(65) sin(cos1(9/3sqrt(65))) / 2 3/sqrt(65) sin(cos1(3/sqrt(65))) / 2 hopefully lol

arijit.mech
 4 years ago
Best ResponseYou've already chosen the best response.0PLEASE SEE THE DRAWING........

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1i got a few more errors, but the concepts seems solid ;)

mathmate
 4 years ago
Best ResponseYou've already chosen the best response.1So 2,3,6 are the intercepts, sqrt(13), sqrt(45), sqrt(40) are the sides.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1right, and you can heron that for sheer simplicity

mathmate
 4 years ago
Best ResponseYou've already chosen the best response.1Some "exact" answers would be nice! :)

mathmate
 4 years ago
Best ResponseYou've already chosen the best response.1Agree, we also want as many different ways as possible.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1dw:1324567464017:dw

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1sqrt(45) sin(y) = height sqrt(13) = base

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1b*h/2 = 3sqrt(14) if i did it right that time

mathmate
 4 years ago
Best ResponseYou've already chosen the best response.1Wow! 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
 4 years ago
Best ResponseYou've already chosen the best response.1the bh/2 is after the vectors give us a sin for an angle to determine the height

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1i 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
 4 years ago
Best ResponseYou've already chosen the best response.2\[A= \frac{1}{2}\sqrt{x^2y^2(x \cdot y)^2}\] with (using amistre's two vectors) x=<2,3,0> y= <0,3,6> gives 0.5sqrt( 13*4581)= 1.5sqrt(659)= 1.5sqrt(56)= 3sqrt(14)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1work in progress here dw:1324568575430:dw \[\int_{0}^{6}distance.x.to.y\ dz\]

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1\[X_i=\left(\frac{z6}{3},0,z\right)\] \[Y_i=\left(0,\frac{z6}{2},z\right)\] \[X_iY_i=\left(\frac{z6}{3},\frac{z+6}{2},0\right)\] distance from X to Y:\[\sqrt{(\frac{z6}{3})^2+(\frac{z+6}{2})^2}\] \[\sqrt{\frac{z^26z+36}{9}+\frac{z^26z+36}{4}}\] \[\sqrt{\frac{4z^24.6z+4.36+9z^29.6z+9.36}{36}}\] \[\frac{\sqrt{13z^278z+468}}{6}\] ergo lol \[\int_{0}^{6} \frac{\sqrt{13z^278z+468}}{6}dz\]

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1hmm, wolf says that about 19, which is a bit off from the 3sqrt(14) i wonder why

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1348 not 468 ... but still thats off

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.1oh well, it was a thought. guess i dont know how to freehand integrals yet :)

mathmate
 4 years ago
Best ResponseYou've already chosen the best response.1OK, 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^2y^2x.y) : phi 5. half magnitude of crossproduct, (1/2)PxQ: mathmate
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