Question

The perimeter of a rectangle is equal to 28 cm. two squares are constructed such that two adjacent sides of the rectangle are each also the side of one of the squares. the combined area of the two squares is 116 cm². find the lengths of the sides of the squares.

Answer 1

Denote by x the length of the side of the first square and by y the length 
of the side of the second square. 
We deduce from the figure the following equations:
$x^2+y^2=116\text{ (The combined area)}\\2(x+y)=28\text{ (The perimeter)}\\\text{Solving the above system:}\\x=14-y\text{ then }(14-y)^2+y^2=116\\\text{Solving the above equation for y we get:}\\\text{ Either }y=4\text{ or }y=10.\\\text{The above values corresponds to}\\\text{ Either }x=14-4=10\text{ or }y=14-10=4.$

Answer: the length of the sides are 4 and 10.