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Math Help - Trigonometry Problem help please

  1. #1
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    Trigonometry Problem help please

    I am really stuck on a trigonometry problem please help if you can.
    ---------------------------
    A circle has a radius of 'r'
    And its center is at 'C'
    The distance from 'A' to 'B' is 'x'
    find 'r' if
    A = 45 degrees and 'x' equals 15

    C
    -|-\
    -|---\
    r|-----\r< C to B is r and C to D is also r
    -|-------\
    -|---------\B< B to A is 15
    -|-----------\
    -|_____45\x < x equals 15 and 45 is Angle A
    D-------------A

    Thats how it is put in the book and The answer is 36 but I do not know how to solve this
    Because r ends up on both sides of the equation.
    if I go
    sin45 = r/r+15 I get
    (r+15)(sin45) = r
    and so on I really could use some help here because I need trig for programming class. thanks
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  2. #2
    Senior Member Educated's Avatar
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    Hello ryogaki.

    Here's what I think the diagram looks like:


    Trigonometry Problem help please-circle-triangle.png

    You do not need to use the sine rule because it is a 45 degree triangle. Since a length is tangent to the circle and another goes through the centre, we know that it is a right angle triangle, and so we can use the pythagorus theorum.

    Length CD = AD because it is a 45 degree triangle.

    Length BC also equals CD and AD because it is the radius of the same circle.

    Let BC = CD = AD = x

    Now using pythagorus theorum: x^2 + x^2 = (\text{length AC})^2

    We know that length AC = (x + 15)

    Therefore the equation becomes: x^2 + x^2 = (x+15)^2

    Now can you carry on from there?
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  3. #3
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    well heres what the picture really looks like.
    I see what your saying about the 45 degree angle but r is one side and only a part of the other. The triangle comes out further on the CA side
    Also if I just do that equation C squared = a squared + b squared it does not give a value for r and the book says that the answer should be 'r = 36'
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  4. #4
    Senior Member Educated's Avatar
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    Have you tried solving the equation of x^2 + x^2 = (x+15)^2?

    In my instance, x = r. So solve for x and you should get your radius.

    EDIT: Howcome the CA side comes out further? A 45 degree right angle triangle has 2 equal lengths.
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  5. #5
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    Thats what I thought too... it should work out like a 45 45 90 triangle.
    The problem is r is a side but then theres an r + 15 so AB can not just be the radius.
    and every way I try to solve this gives me r on both sides...
    I used your way and csc45 and (t) (t) (t times the square root of 2)
    sec of angle C I dont get how to get r to equal 36...
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  6. #6
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    r^2 + r^2 = (r + 15)^2

    2r^2 = ( r+15)^2

    sqrt(2)*r = r+15

    r[sqrt(2) - 1] = 15

    r = 15/[sqrt(2) - 1]

    Now solve for r.
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  7. #7
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    i do not understand...in step 3 you have r on both sides but then in step four the right r is gone. how did that work?
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  8. #8
    Senior Member Educated's Avatar
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    How sa-ri-ga-ma got from step 3 to step 4:

    \sqrt{2} \times r = r+15

    \sqrt{2} \times r -r = r+15 -r (Subtract r from bith sides)

    \sqrt{2} \times r -r = 15

    r(\sqrt{2} -1) = 15 (Factorise the r)
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  9. #9
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    OMG! I have been so messed up on this whole problem for the littlest of reasons...
    I was subtracting r from r [squareroot 2] instead of factoring....that solution does work and it worked on all the other questions like that. I can not believe after all that algebra I would miss something so simple... Thank you Educated and sa-ri-ga-ma very much. That little thing had me going crazy. stick around I might need help with 1+1 next..lol!
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