Saturday, February 11, 2017

Geometry Problem 1313: Regular Decagon, Pentadecagon, Equilateral Triangle, Congruence

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

Geometry Problem 1313: Regular Decagon, Pentadecagon, Equilateral Triangle, Congruence.

6 comments:

  1. Problem 1313
    Is <COB=360/10=36, then <COQ=36/2=18, so <ABC=144.But <CQB=360/15=24 and <MBC=156, so <ABM=360-144-156=60, therefore triangle ABM is equilateral.Similar
    triangle CDN is equilateral (OC=OD, CN=DN) so ON is perpendicular bisector of CD
    then <NOC=36/2=18 or <NOQ=18+18=36.But <CQN=24, <OQC=24/2=12 (OB=OC, QB=QC)
    and <NQO=24+12=36=<NOQ.Therefore NO=QN.
    APOSTOLIS MANOLOUDIS KORYDALLOS PIRAEUS GREECE

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  2. ‹ABM=360°-(144+156)=60°, AB=BM => ∆ABM equilateral
    E,F on CD, BC. ‹EOF=180°-144=36°,
    ‹CQN=180-144=36°, ‹FQC=180-(90+78)=12° => ‹FQN=36° or ∆ONQ isoceles

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  3. Exterior angle of decagon = 360/10 =36
    Exterior angle of pentadecagon = 360/15 =24

    Hence < ABM = 36+24 = 60.
    But AB = BC = BM
    So ∆ ABM is equilateral

    Similarly ∆DCN is equilateral. Hence ∆ OCN ≡ ∆ ODN.

    Therefore < NOC = ½ < COD = ½ X 360/10 = 18. Similarly < COQ = 18 and so <
    NOQ = 36.

    Also < NQO = 1½ X 360/15 = 36

    Therefore ON = QN

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  4. Number of sides of regular polygons which when placed together so that one side is common and forms an angle of 60 (other than 10,15)
    7,42
    8,24
    9,18
    12,12

    ReplyDelete
    Replies
    1. Good info Sumith.

      Any two regular polygons that satisfy the equation 6(n1+n2) = n1*n2 form an angle of 60.

      Where n1 = no of sides of first polygon
      n2 = No of sides of second polygon.

      Delete
    2. Yes the above pairs are the only ones that satisfy the equation 360/n + 360/m = 60 which is equivalent to the equation u have given.

      Those are the only integer solutions

      Delete