You need to solve a system of equations.

Lets start with the perimeter. We know the perimeter of a rectangle is the sum of all the sides and opposite sides are equal.

2l+2w=p (p=perimeter; w=width; l=length)

A rectangle has all right angles, so the length, width, and diagonal will form a right triangle. We will use the Pythagorean Theorem with the diagonal. (only substitute different variables so we have the same variables to work with).

l^2 + w^2 = d^2

SO, your system is:

2l+2w=p

l^2 + w^2 = d^2 Now put in the values you know.

2l+2w=68cm

l^2+w^2=26cm^2 Now we solve the system.

Let's start by isolating one of the variables in the first equation.

2l+2w=68cm Subtract 2w from each side.

2l= 68cm-2w Now divide all terms by 2.

l = 34cm - w Now sub this in the second equation for l.

(34cm - w)^2+w^2=26cm^2 Now solve for w

(34cm - w)(34cm-w)+w^2=26cm^2

1156cm^2-34cm(w)-34cm(w)+w^2+w^2=676cm Now combine like terms (lets drop the cm to avoid confusion)

2w^2--68w+1156 = 676 Subtract 676 from both sides

2w^2-68w+480=0 Now factor and solve. This will give you both the length and width of the triangle. (and, therefore, for your rectangle.) Don't forget to put the cm back in.

(2w-48)(w-10)=0

2w-48=0 w-10=0

2w=48 w=10 cm

w=24cm

Plug these values into the first equation. One for l and one for w. Doesn't matter which is which.

2l+2w=68cm

2(24cm)+2(10 cm)=68cm

48cm+20cm=68cm

68cm=68cm

Your dimensions are 48cm by 20cm.