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Square with Area Equal to a Triangle


Date: 10/23/1999 at 11:53:34
From: Christina
Subject: Areas of triangles and squares

Given an arbitrary triangle, how can you construct a square with the 
same area as the triangle? I just can't seem to figure it out.


Date: 10/27/1999 at 05:06:46
From: Doctor Floor
Subject: Re: Areas of triangles and squares

Hi Christina,

Thanks for your question.

Given any triangle, construct altitude ha perpendicular to side a. 
Then bisect side a.

We now have two segments of lengths a/2 and ha respectively. The 
product of these lengths is equal to the area of the given triangle.

Now create the following situation:

           D
           |                AB = ha     BC = a/2
           |                D is on a circle with AC as diameter
           |                BD is perpendicular to AC
     A-----B-----------C

ADC is a right triangle.

We have BD/AB = BC/BD = tan(<DAC), and thus BD is the geometric mean 
of AB and BC. Or, stated differently, BD = sqrt(AB*BC).

And thus BD is the sidelength of the square you are looking for.

If you need more help, just write us back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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