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Math Help - Regular Heptagon

  1. #1
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    Regular Heptagon

    In a given diagram, ABCDEFG is a regular heptagon. The degree measure of the obtuse angle formed by AE and CG is m/n where m and n are relatively prime positive integers. Find m + n.

    I know that the sum of all the angles is 900 degrees. So, the measure of one interior angle is 128.57 degrees.


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  2. #2
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    Angles in a heptagon

    Hello magentarita
    Quote Originally Posted by magentarita View Post
    In a given diagram, ABCDEFG is a regular heptagon. The degree measure of the obtuse angle formed by AE and CG is m/n where m and n are relatively prime positive integers. Find m + n.

    I know that the sum of all the angles is 900 degrees. So, the measure of one interior angle is 128.57 degrees.
    The exterior angles of any polygon always add up to 360^o. So each exterior angle of a regulaer heptagon is \frac{360^o}{7}.

    Now look at the quadrilateral AEFG. By the symmetry of the heptagon, it is a trapezium with AE parallel to GF. So \angle GAE = exterior angle at G = \frac{360^o}{7}. Similarly \angle AGC = \frac{360^o}{7}.

    Therefore the obtuse angle between AE and CG = \frac{360^o}{7}+\frac{360^o}{7} (Exterior angle of a triangle = sum of interior opposite angles)

    = \frac{720^o}{7}

    \Rightarrow m = 720, n = 7, m+n=727.

    Grandad
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  3. #3
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    great work

    Quote Originally Posted by Grandad View Post
    Hello magentaritaThe exterior angles of any polygon always add up to 360^o. So each exterior angle of a regulaer heptagon is \frac{360^o}{7}.

    Now look at the quadrilateral AEFG. By the symmetry of the heptagon, it is a trapezium with AE parallel to GF. So \angle GAE = exterior angle at G = \frac{360^o}{7}. Similarly \angle AGC = \frac{360^o}{7}.

    Therefore the obtuse angle between AE and CG = \frac{360^o}{7}+\frac{360^o}{7} (Exterior angle of a triangle = sum of interior opposite angles)

    = \frac{720^o}{7}

    \Rightarrow m = 720, n = 7, m+n=727.

    Grandad
    I could have never figured this one out on my own.
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