Friday, May 1, 2009

Problem 287: Regular Octagon, Diagonals

Proposed Problem

Problem 287: Regular Octagon, Diagonals.

See also:
Complete Problem 287: Regular Octagon, Diagonals
Collection of Geometry Problems

Level: High School, SAT Prep, College geometry

Posted by Antonio Gutierrez at 9:17 AM
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9 comments:

  1. AjitMay 2, 2009 at 10:18 PM

    Let G:(0,0) be the origin. Hence, F:(a/V2,a/V2), B:(-a-a/V2,a+a/V2) and A:(-a -a/V2,a/V2) where V=square root. Slope of line BF = (aV2-a-a/V2)/(a/V2+a+a/V2)= 1-V2. Hence, BF:y=(1-V2)x+c and passes thru F:(a/V2,a/V2). On substitution, we obtain c=a. Thus, BF: y =(1-V2)x + a while DG is x=0 or M:(0,a) and MA^2=(a+a/V2)^2+(a-a/V2)^2 = 3a^2. MA = x in the diagram; hence, x=a*V3
    Ajit:ajitathle@gmail.com

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  2. AnonymousOctober 1, 2009 at 5:20 AM

    Since B and F are opposite vertices, BF bisects angle ABC. angle ABM = 67.5.
    join AG observe that AGMB is a parallelogram, angle AGM = 67.5 and GM = AB = a.
    In triangle AGH apply cosine rule to get AG.
    Now AG will be known GM will be Known and also angle AGM is known.
    AM can be found in terms of a using cosine rule.

    ReplyDelete
  3. AnonymousMay 21, 2014 at 10:43 PM

    Easily M is the incenter of tr. DHF, so MG=GH; since GH and GD are perpendicular, HM=a.sqrt{2} and <AHM=90 degs, so from tr. AHM, right-angled at H, AM=a.sqrt{3}.

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    Replies
    1. Antonio GutierrezMay 22, 2014 at 6:14 AM

      To anonymous
      Please give the reason for your statement MG=GH, thanks

      Delete
  4. Sumith PeirisSeptember 21, 2015 at 4:03 AM

    The regular octagon is also cyclic and each side subtends an angle of 22.5 degrees at the other 6 points (135/6). So < MGF = 45 and < MFG = 67.5 hence MG = GF = a

    Hence HGM is isoceles and right and < MHG = 45. So < AHM = 135 - 45 = 90. Now applying Pythagoras to Tr. AHM,

    x^2 = 2a^2 + a^2 = 3a^2 and the result follows

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  5. Sumith PeirisSeptember 21, 2015 at 6:44 PM

    It also follows that HME are collinear

    ReplyDelete
  6. AnonymousApril 3, 2019 at 5:55 PM

    Construct BE meeting DG at X. BXE is congruent to MXE, and both are isosceles right. So DX = XM = asqrt2/2. Now Construct HY such that GHY is 45 and Y is on DG. Because DGFE is an isosceles trapezoid, angle HGM is a right angle, so triangle GHY is isosceles right and GY = a.

    But GD must equal a + a sqrt2, as this is the altitude of the octagon. As DM = asqrt2, YG = a, and D,M,Y and G are collinear by assumption, Y must be concurrent with M. So now it's just a matter of solving right triangles. HM = a sqrt2, so AM = asqrt3. (Once angle GHM was found to be 45, AHM had to be 90.

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  7. Sumith PeirisApril 11, 2019 at 1:26 AM

    Alternatively upon realization that Tr.s AMB & CMB are congruent the answer is straightforward

    ReplyDelete
  8. Sailendra TApril 13, 2019 at 11:48 AM

    It can be observed that MGF is an isosceles with MG=GF=a
    =>MGH is an isosceles right triangle with MH=Sqrt(2).a
    =>AHM is a right triangle with x^2=AH^2+MH^2
    =>x=Sqrt(3)a

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