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Homework Help: Finding Coordinates of last Triangle Vertex

  1. Mar 17, 2012 #1
    I am trying to find the last vertex coordinates of a triangle given that

    Vertex 1 = (2,10)
    Vertex 2 = (3,6)

    Angle at Vertex 1 = 75.9638 degrees
    Angle at Vertex 2 = 70.3462 degrees.

    I have tried using the equations based on the length of each side, as well as using the cos dot product rule to try to find x,y but I cannot seem to .

    I always end up with the same equations using both methods, hence only giving me 1 equation for 2 unknowns.
  2. jcsd
  3. Mar 17, 2012 #2


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    You can use the sine law. The distance between vertices 1 and 2 is [itex]\sqrt{(2- 3)^2+ (10- 6)^2}[/itex][itex]= \sqrt{1+ 16}= \sqrt{17}[/itex].
    Further, the angles must add to 180 degrees so the angle at vertex 3 is [itex]180- 75.9638- 70.3462= 33.69[/itex] degrees.
    So the length, r1, of the side from vertex 1 to vertex 3, the side opposite vertex 2, is given by [itex]r1/sin(70.3462)= \sqrt{17}/sin(33.69)[/itex]. Write the equation of the circle with center at vertex 1 with that radius.
    And the length, r2, of the side from vertex 2 to vertex 3 the side opposite vertex 1, is given by [itex]r2/sin(75.9638)= \sqrt{17}/sin(33.69)[/itex]. Write the equation of the circle with center at vertex 2 with that radius.

    Find the points where those two circles intersect by solving the equations simultaneously. There are two solutions, one on either side of the line from vertex 1 to vertex 2.
  4. Mar 18, 2012 #3
    Thanks for the reply HallsofIvy,

    I end up with only 2 complex roots for some reason when i use these 2 equations

    (x-3)^2 + (y - 6)^2 = 52 =(r2^2)
    (x-2)^2 + (y-10)^2 = 49 = (r1^2)

    i got 10x^2 - 284x +2652 = 0 after substitution.

    the answer i am suppose to get for the last coordinate is (9,10)
  5. Mar 18, 2012 #4
    oops my mistske, i did some calculation errors.

    hmm is there anyway to pick the right coordinates when i end up with 2 solutions?

    Both will have the same angle between the vectors , so theres no way to tell which is the coordinate im looking for?
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