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Math Help - Help With a Diagram

  1. #1
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    Help With a Diagram

    Can someone help me solve the following diagram, if at all possible...



    Solve for Angle: Theta

    Regards,
    - Mark
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  2. #2
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    Re: Help With a Diagram

    If we call the hypotenuse of the smaller triangle \displaystyle \begin{align*} x \end{align*} , then our large triangle has dimensions \displaystyle \begin{align*} A, B + x, \sqrt{A^2 + (B + x)^2} \end{align*} (by Pythagoras).

    Since the triangles are similar, their corresponding sides are in the same proportion, so

    \displaystyle \begin{align*} \frac{A}{R} &= \frac{\sqrt{A^2 + (B + x)^2}}{x} \\ \frac{A^2}{R^2} &= \frac{A^2 + (B + x)^2}{x^2} \\ \frac{A^2}{R^2} &= \frac{A^2 + B^2 + 2Bx + x^2}{x^2} \\ A^2x^2 &= R^2\left(A^2 + B^2 + 2Bx + x^2\right) \\ A^2x^2 &= A^2R^2 + B^2R^2 + 2BR^2x + R^2x^2 \\ \left(A^2 - R^2\right)x^2 - 2BR^2x &= A^2R^2 + B^2R^2 \\ x^2 - \frac{2BR^2}{A^2 - R^2}x &= \frac{A^2R^2 + B^2R^2}{A^2 - R^2} \\ x^2 - \frac{2BR^2}{A^2 - R^2}x + \left(-\frac{BR^2}{A^2 - R^2}\right)^2 &= \frac{A^2R^2 + B^2R^2}{A^2 - R^2} + \left(-\frac{BR^2}{A^2 - R^2}\right)^2 \\ \left(x - \frac{BR^2}{A^2 - R^2}\right)^2 &= \frac{\left(A^2R^2 + B^2R^2\right)\left(A^2 - R^2\right) + B^2R^4}{\left(A^2 - R^2\right)^2} \\ \left(x - \frac{BR^2}{A^2 - R^2}\right)^2 &= \frac{A^4R^2 - A^2R^4 + A^2B^2R^2 - B^2R^4 + B^2R^4}{\left(A^2 - R^2\right)^2} \\ \left(x - \frac{BR^2}{A^2 - R^2}\right)^2 &= \frac{A^4R^2 - A^2R^4 + A^2B^2R^2}{\left(A^2 - R^2\right)^2}\end{align*}

    \displaystyle \begin{align*} x - \frac{BR^2}{A^2 - R^2} &= \pm \frac{\sqrt{A^4R^2 - A^2R^4 + A^2B^2R^2}}{A^2 - R^2} \\ x &= \frac{BR^2 \pm \sqrt{A^4R^2 - A^2R^4 + A^2B^2R^2}}{A^2 - R^2} \end{align*}


    Therefore, the height of the larger triangle is \displaystyle \begin{align*} B + \frac{BR^2 \pm \sqrt{A^4R^2 - A^2R^4 + A^2B^2R^2}}{A^2 - R^2} \end{align*} , and so we have the relationship

    \displaystyle \begin{align*} \tan{\theta} &= \frac{B + \frac{BR^2 \pm \sqrt{A^4R^2 - A^2R^4 + A^2B^2R^2}}{A^2 - R^2}}{A} \\ \theta &= \tan^{-1}{\left(\frac{B + \frac{BR^2 \pm \sqrt{A^4R^2 - A^2R^4 + A^2B^2R^2}}{A^2 - R^2}}{A}\right)} \end{align*}
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  3. #3
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    Re: Help With a Diagram

    Here's an alternative method.

    First some construction. (Apologies, I don't know how to incorporate diagrams, maybe one of the sites resident experts could suggest a method ?)

    Run a line from the centre of the circle parallel with the hypotenuse down to the base of the triangle. Call the three points along the base D, E and F. (D is at the end of the hypotenuse, E is the new point and F is at the rightangle). Run another line from E upto the hypotenuse to meet it at rightangles (it will be parallel with the line in the circle), and call this point G.

    In the new triangle DEG, GE/DE=\sin\theta, but GE=R, so DE=R/\sin\theta,

    in which case EF=A-R/\sin\theta.

    Therefore, \frac{B}{A-\frac{R}{\sin\theta}}=\tan\theta=\frac{sin\theta}{  \cos\theta}.

    Mutiply top and bottom of the LHS by \sin\theta to get \frac{B\sin\theta}{A\sin\theta-R}=\frac{\sin\theta}{\cos\theta},

    cancel the \sin\theta 's, cross multiply, rearrange, and you arrive at A\sin\theta-B\cos\theta=R.

    From hereon it's routine, put the LHS equal to K\sin(\theta-\alpha), expand and equate with the existing LHS to get

    K\cos\alpha = A,\quad K\sin\alpha=B so that K=\sqrt{A^{2}+B^{2}} and \tan\alpha=B/A.

    Finally then, R=\sqrt{A^{2}+B^{2}}\sin(\theta-\alpha),

    \theta=\arcsin\frac{R}{\sqrt{A^{2}+B^{2}}}+\arctan  \frac{B}{A}.

    S*** what an idiot !!!

    Draw a line from the centre of the circle down to the base of the hypotenuse. The "new" hypotenuse will be of length \sqrt{A^{2}+B^{2}}.
    The angle \theta will be the sum of those two angles which will be, from the two triangles \arcsin\frac{R}{\sqrt{A^{2}+B^{2}}} and \arctan\frac{B}{A}.
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  4. #4
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    Re: Help With a Diagram

    Draw the segment containing the two right angles. We'll call it PQ, where P is the right angle in the smaller triangle and Q is the right angle in the larger triangle. Suppose the three vertices of the large right triangle are, in clockwise order, QRS, where Q is a right angle.


    The hypotenuse of the large triangle is given by RS = \frac{A}{\cos \theta}. Also, PS = R \tan \theta.

    Therefore, RP = RS - PS = \frac{A}{\cos \theta} - R \tan \theta = \frac{A - R \sin \theta}{\cos \theta}

    Applying the law of cosines,

    PQ^2 = R^2 + B^2 - 2RB \cos (180 - \theta)

    PQ^2 = A^2 + (\frac{A - R \sin \theta}{\cos \theta})^2 - 2A(\frac{A - R \sin \theta}{\cos \theta}) \cos \theta

    Equate these two, noting that \cos (180 - \theta) = -\cos \theta

    R^2 + B^2 + 2RB \cos \theta = A^2 + (\frac{A - R \sin \theta}{\cos \theta})^2 - 2A(\frac{A - R \sin \theta}{\cos \theta}) \cos \theta

    And so on.
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