Here's a diagram representing the given information:


Let x = the length of the base
Let y = the height

Given: The area of rectangle ABCD is 12
Since the area of a rectangle = (base)(height), we can write xy = 12

Also, since BD is the hypotenuse of right triangle ABD, we can apply the Pythagorean theorem to write: x² + y² = 5²
Evaluate to get: x² + y² = 25

We now have the following system of equations:
xy = 12
x² + y² = 25

Important: Notice that some of the terms in the above system closely resemble the special product x² + 2xy + y², which can be conveniently factored to become (x + y)².
We're going to use this property to answer our question.

Take the top equation (of our system), and multiply both sides by 2 to get:
2xy = 24
x² + y² = 25

Now add the two equations to get: x² + 2xy + y² = 49
Factor the left side to get: (x + y)² = 49
Take the square root of both sides to get: x + y = 7

Since the perimeter of the rectangle = 2x + 2y, we can take x + y = 7 and multiply both sides of the equation by 2 to get: 2x + 2y = 14
So, the perimeter of the rectangle = 14

Answer: 14