The answer to problem as stated is no. However there is a
similar problem to which the answer is yes. First the problem
as stated is no:
This figure below is drawn to scale. The longer side of the
large rectangle is times the shorter
side.
All golden rectangles are similar. So the answer is
no, because neither of those two rectangles is similar
to the large rectangle.
-------------------------------
Now here is a similar problem where the answer is yes.
We draw the line different:
Take a Golden Rectangle and draw the largest circle
inside it that touches three sides. If we draw a line
parallel to the shorter side and tangent to the circle
and then cut the rectangle into two pieces along that
line, will either of the two smaller rectangles be a
Golden Rectangle?
The big rectangle below is Golden. The short side is "a"
and the long side is . The circle is inscribed
in a square. The bottom side of the square is "a" units
So the short side of the small rectangle on the right is
|------------------------------------------|
Now we will show that the small rectangle on the right is golden.
The ratio of the sides is
Divide numerator and denominator by a
Substitute
Multiply top and bottom by 2
Rationalize the denominator by multiplying by
Divide top and bottom by -2:
Which is , the golden ratio.
So for this problem, the answer is
Yes, the small rectangle on the right is golden.
[I have a hunch this was the problem that was intended.
Point this out to your teacher.]
Edwin