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Hexagon Sides, Length of a Beam

Date: 1 Feb 1995 20:27:58 -0500
From: Cheryl A. Backman
Subject: math q

Hi again. I am totally stumped on these two word problems.  They are 
driving me crazy!  Here goes:

  1.  Determine the length of each side of a regular hexagon if opposite 
       sides are 18 in. apart.

  2.  A wooden board is placed so that it leans against a loading dock 
      to provide a ramp.  The board is supported by a metal beam perpendicular
      to the ramp and placed on a 1 ft. tall support.  The base of the support
      is 7 feet from the point where the ramp meets the ground.  The slope of
      the ramp is 2/5.  Find the length of the beam to the nearest hundredth
      of a foot.

Please help! Could you possibly respond by 11p.m. this evening or 6:30 
tomorrow morning? 


Date: 3 Feb 1995 18:22:35 GMT
From: Dr. Math
Subject: Re: math q

Dear Michele,

Sorry we weren't able to respond as quickly as you requested -- we are
pretty busy  right now, so it is taking us a few days to answer
questions.  Let's look at your questions now, though -- better late than
never, eh?

1.  A regular hexagon has 6  equal sides that are at 120 degree angles
with one another.  To do your problem, I think it helps most to draw a
picture.  Since I am not very good at drawing on the computer, you'll have
to draw your own picture on a piece of paper.  Okay, draw a regular
hexagon.  Now, find a vertex in the hexagon (it doesn't matter  which one,
since the hexagon is regular) and call it point B.  Find the points
corresponding to the vertices that are next to this vertex and call them A
and C.  Then, draw a line connecting A and C-- you should now have a
triangle with a 120 degree angle at point B.  The length of the side of
the triangle opposite point B is going to be 18 inches.  Can you see why
from your picture?  The line from A  to C  is the line connected opposite
sides.  So, now, draw a line from B to the middle of line AC so you have
two identical right triangles.  Call this new vertex D.  Now look at
triangle ABD.  The angle at vertex B is half of what it was before we
drew the line to D, so it is now 60 degrees, while the angle at D is 90
degrees.  Since the interior angles must add up to 180 degrees, angle A
must be 30 degrees.  This makes things pretty nice because 30-60-90
triangles have some nice properties.  In particular, in 30-60-90 
triangles, the side opposite the 90 degree angle is 2/sqrt 3 times as
long as the side opposite the 60 degree angle.  That means that in our
picture, side AB is 2/sqrt 3 times as long as side AD.   In our picture,
side AD is half as long as side AC which was 18", so side AD is 9". 
This means side AB has length (9")2/sqrt 3.  Side AB is one of the sides
of the regular hexagon, so the regular hexagon that has sides 18" apart
has sides of length 18"/sqrt 3. 

2.  I'm not completely clear on the wording of this problem, so maybe you
could write back and tell me if I am understanding it correctly.  I drew a
picture, and here is roughly what it looks like.  Is  it right? 

            /90\  x
           /    \    
          /      \    |
         / -7ft-    1 ft.

I know the picture is not in correct  proportions.  The 90 indicates a 90
degree angle, the -7ft- means the horizontal distance on the ground there
is 7ft, and the    |
     1 ft    means 1 ft in between the ground and the beam

You are trying to find the distance x, given that the line is rising with
a slope of 2/5, right?  Write me back to tell me if I have the problem
right and I will help you work through it, okay?   

Hope this helps.

--Sydney, "dr. math"
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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