I need help: (if a video could be provided that would be awesome)

Question

556.A rectangle is 2 inches wide, and more than 2 inches long. It so happens that this rectangle can be divided, by a single cut, into a 2-inch square and a small rectangle that has exactly the same shape as the large rectangle. What is the length of the large rectangle?

Answer 1

I'm new and don't yet know how to post videos, pictures, equations, etc.so I'll explain.

The original rectangle had a long side of unknown length x, with x>2, and a short side of 2.

If you slice off a square of 2 by 2, you are left with sides for the small rectangle of x-2 and 2.

It is impossible for the long side of the small rectangle to be x-2, which would make the short side be 2. This is because the rectangles can't have the same shape with the same short side. Thus the long side of the small rectangle is 2 and the short side of the small rectangle is x-2. So.....

Original rectangle long side to short side ratio must be equal to small rectangle long side to short side ratio.

Set x/2 =2/(x-2), cross multiply and get (x)(x-2) = 4, or x²-2x-4=0.Solve using the quadratic formula with a=1, b=(-2),and c=(-4). Get x=1+√5=3.24. This is the long side of the original rectangle. The short side is 2.

For the small rectangle, the long side is 2 and the short side is x-2 = 3.24-2=1.24

Sorry for all the verbiage. I need to figure out how to add equations.

Answer 2

(x - 2)/2 = 2/x; x² - 2x = 4; x² - 2x - 4 = 0; x = 1 + √5