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Triangle Donut (Posted on 2006-01-01) Difficulty: 2 of 5
If X is the area of any triangle-PQR and you make a hole in the center of this triangle by first joining the midpoints of each side to create smaller triangle-STU and then again joining the midpoints of triangle-STU to create the hole in the center of this triangle, it would create a donut something like this:
         / \
        /   \
       /     \
    S /_______\ U
     / \  A  / \
    /   \/_\/   \
   /     \ /     \
P /_______V_______\ R
What is the area of the triangle donut in terms of X?

See The Solution Submitted by tanx    
Rating: 3.0000 (1 votes)

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Solution My interpretation | Comment 2 of 14 |

I think we'd all agree that triangle STU has 1/4 the area of triangle PQR; that is, it has area X/4.  Also, the smallest triangle has an area 1/4 of that, or X/16.

So the hole has area X/16; but what is the non-hole part of the donut?  Is it all of PQR except for the hole, with triangle STU being a mere aid in construction?  Then the area of the donut is 15 X / 16.  But if the outer boundary of the donut is triangle STU, then the area of the donut is 3 X / 16.

  Posted by Charlie on 2006-01-01 12:53:42
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