How Can Triangle Coordinates Be Found with Given Information?

[super15mn]

so i have a problem, and I am not sure it can be solved here but i thought maybe someone could point me in the right direction. I have 2 sets of geographical coordinates, these coordinates are 0.87 miles apart, the unknown here is a third set of coordinates.. now, i know point A to point B is 0.87 miles and the distance from point A to unknown point C is .3696969 miles and the distance from point B to unknown point C is 0.61 miles...this helps if you draw a triangle. can the coordinates to unkown C be found with the given information?

please email me if you know anything about how i can solve this..

thanks

 

[Data]

There will be more than one possible point (in particular, there will be two of them), and any answer will be relative to the positions of $A$ and $B$, but let's take a look:

Set A = (0,0). B is 0.87 miles away, so set B = (0.87, 0). We need a point C = (x, y) s.t.

$$d(C,A) = \sqrt{x^2 + y^2} \approx 0.37$$

and

$$d(C,B) = \sqrt{(x-0.87)^2 + y^2} \approx 0.61.$$

Subtracting the square of the first equation from the square of the second yields

$$-2(0.87)x + 0.87^2 = (x-0.87)^2 - x^2 = 0.61^2 - 0.37^2,$$

which results in $x\approx 0.30$. Now plug that back into the square of the first equation, to get

$$0.30^2 + y^2 = 0.37^2,$$

and solving for $y$ yields $y \approx \pm 0.217$.

So C is approx. either (0.30, 0.217) or (0.30, -0.217). In other words, if B is exactly east of A, then C is 0.30 miles east of A and 0.217 miles either north or south of A.

 

[super15mn]

thanks so much for the help

is there anyway we can translate this to coordinates,

A= N 35 08.487, W089 52.046
B= N 35 09.025, W089 52.692

would that involve an algorithm, i have some general direction from point A to unknown point C, is definately West of point A and more than likely north.

Point A to Point B is NorthWest in direction

 

[Data]

I can try to help you, but I'm afraid I'll need some help first: Are those coords latitudes/longitudes (if so, I assume degrees and minutes?)? I really don't know anything about the standard notation for coordinates used in practice!

If it helps: Starting at A, C is 0.30 miles in the direction of B and 0.217 miles perpendicular to the direction from A to B (there are two directions perpendicular to the direction of B. There's not enough info to tell which is the right one. However, once I figure out the coordinates you gave me I can probably tell you which one )

Edit: Ok, here's a further best guess; If I understand your coordinates properly then B is northwest of A. If C is also northwest of A, I'm going to assume that you mean it's also probably northeast of the line joining A to B.

In that is the case, if you start at A, then you should walk 0.30 miles directly towards B, turn right $90^\circ$, and walk 0.217 miles that way.

Give me a few minutes and I should be able to get you proper longitude/latitude coords.

 

[super15mn]

they are latitude and longitude, the lat is the North and the Long is the West. there are 3 different system od coord, the ones i use are the WGS84 system, http://en.wikipedia.org/wiki/WGS84 these coords are DDD MM.MMM where D is degrees and M is minutes.

 

[super15mn]

i believe it will be to the south and to the west, becausr my "directions" say i will need to start out waling northwest and suddenly follow a straight line south west

 

[Data]

Alright, after a bit of calculation then:

C should lie at (approximately) either

N 35 09.010, W 89 52.143

or

N 35 08.576, W 89 52.683.

Based on your info above it seems as though the first one is more likely.

Edit: Okay, based on your last post the second one's seems like it's the right one . You can probably make a guess as to which is more likely from the coords.

Edit 2: Whoops, wait a minute. I swapped the distance between C and B with the distance between C and A in my calcs. Just a second!

 

[super15mn]

ok when i looked at these coords it gave me approximately 0.61 miles from A to C, and Ac to C needs to be 0.36969, 0.61 is the Distance from B to C?

 

[super15mn]

you are really good, you realized it was wrong before i did!

 

[Data]

Yep, see above . Here are the corrected (approx.) coords:

N 35 8.541, W 89 52.432

N 35 8.804, W 89 52.105

you are really good, you realized it was wrong before i did!

Wouldn't be much help if I gave you wrong directions!

 

[super15mn]

how is this done? i really only need it for this one time but, is your math really complex?

 

[Data]

This is a pretty simple problem, but that's mitigated by the fact that I'm pretty rusty at plane geometry! It's also complicated by not really being plane geometry at all. In fact one of the approximations in my calculations was that I could approximate the Earth by as flat on small distance scales (the spherical geometry comes in because the distance that $1^\circ$ of longitude represents changes based on the latitude).

Past that it's just a bunch of simple trig. I would show you the steps, but it'd be fairly incomprehensible without diagrams.

 

[super15mn]

well i appreciate it, i am a chemistry grad, and now in pharmacy school, this seemed WAY over my head thanks for the help

 

[Data]

No problem!