+312 A large square is divided into 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?
+128406 The side of the large square = 5
The side of the small square = √7
Let the smaller dimension of a ectangle = x
Let the larger dimension of a rectangle = y
And we have this system
2x + √7 = 5 ⇒ x = ( 5 - √7) / 2 (1)
x + y = 5 (2)
Sub (1) into (2)
(5 - √7) / 2 + y = 5
y = 5 - (5 - √7)/2 = 5/2 + √7/2 = ( 5 + √2)/ 2
The square of the diagonal length =
x^2 + y^2 =
(5 - √7)^2 / 4 + ( 5 + √7)^2/4 =
[ 25 - 10√7 + 7 + 25 + 10√7 + 7 ] / 4 =
[ 50 + 14 ] / 4 =
64/ 4 =
16
So....the length of the diagonal of a rectangle = √16 = 4
