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Math Help - Finding length of sides of triangles using differences in distance to a common point

  1. #1
    DSG
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    Finding length of sides of triangles using differences in distance to a common point

    I have three co-linear points A, C, & B respectively. The distance between each of the points is known.

    I have a point P at an unknown location, perhaps on the same line if possible, but usually always not co-linear with ACB.

    The distance from P to C, called 'c', is unknown.

    The distance from P to A is a + c, where 'a' is known (and 'c' is still the unknown distance from P to C). I.E., the distance from P to A is 'a' further than 'c' from P to C.

    The distance from P to B is b + c, where 'b' is known (and 'c' is unknown). I.E., the same story as with point A.

    a and b can be negative, but a+c and b+c are > 0 of course.

    What is the distance from P to C ('c')?

    Part II
    Let's say the distance from P to A is unknown, but the difference to C and the difference to B are known. What is the distance to A?

    Thanks,
    Dale
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  2. #2
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    Finding length of sides of triangles using differences in distance to a common point-capture.jpg

    using the law of cosines we can write 2 equations.

    let the distance from AC=x and CB=y and these are known.

    E_1: (a+c)^2=c^2+x^2-2cx\cos(\theta)

    (b+c)^2=c^2+y^2-2cy\cos(\pi-\theta)

    using the subtaction Identity for cos gives

    (b+c)^2=c^2+y^2-2cy( \cos( \pi ) \cos( \theta )+\sin( \pi )\sin(\theta))

    this finally gives us

    E_2: (b+c)^2=c^2+y^2+2cy\cos(\theta)

    E_1 and E_2 give a system of equations with the only unknowns c and theta...

    I hope this helps
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  3. #3
    DSG
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    Thanks for the help but I think I need a little more help on this. I do not know the value of theta.

    All I know is the distance between points A, C, & B and I know that from point P, A is 'a' further from P than C is from P and that B is 'b' further from P than C is from P.

    Thanks,
    Dale
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by DSG View Post
    Thanks for the help but I think I need a little more help on this. I do not know the value of theta.

    All I know is the distance between points A, C, & B and I know that from point P, A is 'a' further from P than C is from P and that B is 'b' further from P than C is from P.

    Thanks,
    Dale

    You could isolate c in the first equation and sub it into the second equation and then solve for theta.

    I didn't do the work by hand but maple gives the solution.

    Finding length of sides of triangles using differences in distance to a common point-capture.jpg

    Good luck.
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  5. #5
    DSG
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    Thanks for your help. With the equation solved for c, I presume I do not need the equation for theta (since I do not care about the angle; I only need the distance from P to A, B, & C).

    Thanks,
    Dale
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  6. #6
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    That's correct. You have 2 equations with 2 unknowns. Using substitution it is possible to eliminate 1 unknown to establish a formula for the other unknown; in this case the desired c.
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  7. #7
    DSG
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    The Empty Set and I Know One,

    I appreciate your help.

    Thanks,
    Dale
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  8. #8
    DSG
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    I realize this makes it into a different problem, but what if A, B, and C are NOT co-linear, but instead form a triangle (actually, the base of a three-sided pyramid)? Again, the distances between A, B, and C are known.
    Thanks,
    Dale
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