Question

What is the area of the triangle formed by connecting the points (0, 0, 0), (1, 1, 2), and (3, 4, 2)?

Answer 1

I am thinking heron's formula, but it would be a large number involving many square roots...

Answer 2

Well I would use the vector product and divide the result by two.

Answer 3

For instance you can setup two vectors, the origin and the point (1,1,2) and the origin and the point (3,4,2)

Answer 4

as @waterineyes, you will get the area of a Parallelogram, but that's great because if you divide that by 2 you'll do fine.

Answer 5

I have not done vectors yet...

Answer 6

Then you have to go with Herons formula..

Answer 7

hmm okay, that's one 'easy' way I could suggest, I wouldn't have another idea as of now, because this is 3 Dimensional.

Answer 8

Alright. Let's see where that leads...

Answer 9

Firstly calculate the distance of all the sides..

Answer 10

They are \(\sqrt{6}\), \(\sqrt{13}\), and \(\sqrt{29}\).

Answer 11

*lengths..

Answer 12

You have to take it in decimal forms..

Because you are now to find Semi perimter..

Answer 13

Also no calculators...I'm going to try something else for a second...

Answer 14

|3 4 2|
|1 1 2|
|0 0 0|
Is there a way to do it with the way you do coordinate geometry?

Answer 15

I think so but not sure..

Answer 16

\[\tiny\sqrt{\frac{\sqrt{6} + \sqrt{13} + \sqrt{29}}{2}(\frac{\sqrt{6} + \sqrt{13} + \sqrt{29}}{2} - \sqrt{6})(\frac{\sqrt{6} + \sqrt{13} + \sqrt{29}}{2} - \sqrt{13})(\frac{\sqrt{6} + \sqrt{13} + \sqrt{29}}{2} - \sqrt{29}}\]That's very ugly :/

Answer 17

Ha ha ha..

Take them in decimals form..

Answer 18

lol...alright...I'll break the rules until I learn vectors :/

Answer 19

s ≈ 5.72

Answer 20

Now use the heron's formula for Area..

Answer 21

a ≈ 2.45
b ≈ 3.6
c ≈ 5.385
3.27(5.72)(2.12)(0.335) = 13.28?

Answer 22

I have not checked it if your calculations are right or not..

Answer 23

Let me check..

Answer 24

http://www.wolframalpha.com/input/?i=Area+%280%2C+0%2C+0%29+%281%2C+1%2C+2%29+%283%2C+4%2C+2%29
Wolfram doesn't agree with me...

Answer 25

There is one square root outside too I guess..

Answer 26

Oh yeah :/
It works now :)

Answer 27

THank you! I'm guessing that if I knew vectors it would be a lot easier?

Answer 28

Just wait..

It will be easy when you will study about vectors..

Welcome dear..