Let R be the area of rectangle ABCD and
T the area of triangle APQ (where P and
Q are points on sides BC and CD of ABCD respectively).

What is the minimal value of |BP|+|DQ|
in terms of R and T?

Note: R and T are constants with 0 < T ≤ R/2.
Therefore, |BP| and |DQ| are not independent variables
(i.e. if P varies between B and C, then Q
must vary between C and D such that T
stays constant).  

AB = a

BC = b

BP = x

QD = y

Area of rectangle = area of component triangles

R = T + 1/2 * ( ax + by + (a-y)(b-x))

and since ab = R

xy = R - 2T

Then x + y = x + ( R - 2T )/x

which is minimum when x + y = sqrt( R - 2T )

Posted by xdog on 2010-10-14 13:28:29