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Math Help - Finding distance between two points using two angles

  1. #1
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    Finding distance between two points using two angles



    Hello, I'm looking for help in learning to solve this kind of problem. I want to know the distance between point A and point B, the length of X, and the length of Y (assuming that X and Y intersect at a right angle), using the information I have to start with (everything in blue).

    If anybody can make sense of this, please help! I'm stumped, I'm an absolute idiot in mathematics (especially trigonometry) but I'm pretty sure I can follow along if the steps are basic enough.

    Thanks in advance!
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Is there no more information given, such as a right angle, parallel lines?

    Also, from what I see, 10 and 5 are lengths, right?
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  3. #3
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    All of the information given is above: the numbers in black are lengths (you are correct), both of the length 5 lines are at 90 degrees with line X, I'm sorry I forgot to mention that.
    Last edited by Choscura; January 18th 2011 at 08:59 AM.
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  4. #4
    MHF Contributor Unknown008's Avatar
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    I tried to get AB but I didn't find much, however, I found another method, involving putting this on a cartesian plane, and solving simultaneously to get the coordinates of B. Knowing the coordinates of A, AB can then easily be obtained. How about it? Do you think you can use that?
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  5. #5
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    yes, that would work, I think. Looking at it it looks like the 5" line across X from Y is directly proportional to Y when the two lines from that 5" line intersect the ends of Y forming a proportional triangle- the tricky bit is finding the size of the proportions (if this makes sense....)

    I'm still absolutely stumped by this, I can look at it and feel the answer staring me back in the face but I can't figure out how to get something that works at all, much less every single time. Please, post the cartesian coordinate solution and maybe this can finally make sense.
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  6. #6
    MHF Contributor Unknown008's Avatar
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    Okay, put the whole thing on cartesian plane.

    You'll notice that the 10, 10, 10 triangle at the lower left is an equilateral.

    Put the origin in the middle of the base of this triangle. A then becomes point (2.5, 2.5\sqrt3) (see if you can get that)

    Then, at coordinates (5, 0), there is this line going up to B. We can get the gradient of this line, knowing that:

    \tan70 = m_1

    Hence, we get the equation of this line, let's name it line 1.

    Now, there is another line from the vertex of the equilateral to B. That vertex has coordinates (0, 5\sqrt3)

    The gradient of line 2 is \tan(90 - (50+30)) = m_2 (can you see how?)

    You have a gradient and a point, you get the equation of line 2.

    Equate both lines, to get the point B.

    You know point B and point A, you can find AB.
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  7. #7
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    Quote Originally Posted by Choscura View Post

    Hello, I'm looking for help in learning to solve this kind of problem. I want to know the distance between point A and point B, the length of X, and the length of Y (assuming that X and Y intersect at a right angle), using the information I have to start with (everything in blue).

    If anybody can make sense of this, please help! I'm stumped, I'm an absolute idiot in mathematics (especially trigonometry) but I'm pretty sure I can follow along if the steps are basic enough.

    Thanks in advance!
    1. I've modified you drawing a little bit. (see attachment)

    2. The grey triangle EFG is an equilateral triangle with the side length 10. Thus the 3 interior angles are 60.

    3. From this you can determine the interior angles of the triangle GFB (values in green). Use the Sine rule to determine |\overline{GB}| and |\overline{FB}|.

    4. AB is a median of the triangle FGB. Use Cosine rule to determine it's length.
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