A large square is divided into 4 small congruent rectangles and a small square as shown. The areas of the large and small squares are 25 and 7, respectively. What is the length of a diagonal of a small rectangle?
The side of the large square = sqrt (25) = 5
The side of the small square = sqrt (7)
The smaller dimension of a rectangle, d, can be found as
2d + sqrt (7) = 5
2d = 5 - sqrt (7)
d = [ 5 - sqrt (7) ] / 2 = (5/2) - sqrt (7) / 2
So the larger dimension, D = 5 - [ 5/2 - sqrt(7) / 2 ] = 5/2 + sqrt (7) / 2
So.....the length of the diagonal of a rectangle =
sqrt [ d^2 + D^2 ] =
sqrt [ ( 5/2 - sqrt(7) / 2 )^2 + ( 5/2 + sqrt (7)/2)^2 ] =
sqrt [ 25/4 + 7/4 + 25/4 + 7/4] =
sqrt [ 64/4] =
sqrt [ 16] =
4