If you call the width x then the length is x + 7.
You use pythagoras to find the diagonal, so
x² + (x + 7)² = 13² -> x² + x² +14x +49 = 169
2x² + 14x -120 = 0 -> x² + 7x -60 =0
(x+12)(x-5) = 0 in this case x can't be negative (it's a measure) so x = 5
dimensions are 5cm and 5+7=12cm

or you might know 5,12,13 as a pythagoras triple which would avoid all the working!

A frame has a right angle so we will use
Pythagorean theorem
c^2 = a^2+b^2
or for this diagonal and frame
d^2 = L^2+w^2
It says the length is 7 inches longer than the width
so
L= w+7
plugging in the measure of the diagonal and the length
in terms of the width we get
13^2 = (w+7)^2+w^2
FOILing and raising 13 to the send we get
169 = w^2+14w+49+w^2
Because we have a w^2 we move ever term to one side
setting the equation to zero
0 = 2w^2+14w-120
If I divide ever term by the common factor of 2
I get
0 = w^2 +7w-60
factoring
0=(w+12)(w-5)
so I get widths possibly
-12 inches or 5 inches
I chose the positive units
then add 7 inches to the 5 to get the length
of 12 inches
so
width is 5 inches
length is 12 inches