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?
Since S,T,U are the mid points of ¥ÄPQR so ST = ¨öQR and UT = ¨öPQ and SU = ¨öPR and all sides ST, TU, US are parellel to RQ, QP and PR respectively.
Therefore area of ¥ÄSTU = ¨ù area of ¥ÄPQR
Let A be the mid-point of ST,B be the mid-point of TU, and C be the mid-point of US.
SO similiarly as above
area of ¥ÄABC = ¨ù area of ¥ÄSTU
Therefore area of ¥ÄABC = (1/16)*X
and area of triangle donut = (1/4)*X-(1/16)*X = (3/16)*X
Posted by akash
on 2006-01-01 11:16:26