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Math Help - Maximum Triangles

  1. #1
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    Maximum Triangles

    What is the maximum number of distinct triangles that can be formed if m<A = 30, b = 8, and a = 5
    (1) 1 (3) 3
    (2) 2 (4) 0



    Can I use law of sines to answer such questions about how many different triangles?
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  2. #2
    A riddle wrapped in an enigma
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    Quote Originally Posted by magentarita View Post
    What is the maximum number of distinct triangles that can be formed if m<A = 30, b = 8, and a = 5
    (1) 1 (3) 3
    (2) 2 (4) 0


    Can I use law of sines to answer such questions about how many different triangles?
    Hello Magentarita,

    Let's look at both cases: Recall that the sine of an angle is equal to the sine of its supplement.

    Case I (diagram 1)

    \frac{\sin A}{A}=\frac{\sin B}{B}

    \frac{\sin 30}{5}=\frac{\sin B}{8}

    \sin B = .8

    B \approx 53.1^{\circ}


    Case II (diagram 2)

    \triangle B'BC is isosceles.

    \angle B is supplementary to \angle AB'C

    \angle AB'C=180-53.1 \Rightarrow 126.9

    \frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}

    So, looks like 2 triangles can have the characteristics you describe.

    \triangle ABC \ \ and \ \ \triangle AB'C
    Attached Thumbnails Attached Thumbnails Maximum Triangles-diagram1.jpg   Maximum Triangles-diagram2.jpg  
    Last edited by masters; January 5th 2009 at 12:53 PM.
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  3. #3
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    Tell me......

    Quote Originally Posted by masters View Post
    Hello Magentarita,

    Let's look at both cases: Recall that the sine of an angle is equal to the sine of its supplement.

    Case I (diagram 1)

    \frac{\sin A}{A}=\frac{\sin B}{B}

    \frac{\sin 30}{5}=\frac{\sin B}{8}

    \sin B = .8

    B \approx 53.1^{\circ}


    Case II (diagram 2)

    \triangle B'BC is isosceles.

    \angle B is supplementary to \angle AB'C

    \angle AB'C=180-53.1 \Rightarrow 126.9

    \frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}

    So, looks like 2 triangles can have the characteristics you describe.

    \triangle ABC \ \ and \ \ \triangle AB'C
    Is this the ambiguous case?
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by magentarita View Post
    Is this the ambiguous case?
    Yes, it is.
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  5. #5
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    by the way...

    Quote Originally Posted by masters View Post
    Yes, it is.
    By the way, I like playing with ambiguous case trig questions.
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