Condition for Rectangle and Square to have the same area

Question

Prove that if a rectangle and a square has the same area if and only if the length of the square is the geometric mean of the side length of the rectangle.

I'm not sure how to start this problem. I tried cutting them into triangle without any success.

Answer 1

Hint: The geometric mean of two numbers a and b is $\sqrt{ab}$.

What happens when you equate areas of square of side $a$ and rectangle of length $l$ and breadth $b$?

Answer 2

We know that geometric mean is $\sqrt{ab}$ And its given that the area will be equal iff geometric mean of the sides of the rectangle equal the side of the square.

So equating what we know till now we get :$$\sqrt{ab}=a$$$$ab=a^2$$ Which implies $a=b$.

I think you can carry on after this