0

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.

Jesse P Francis
  • 1,469
  • 3
  • 27
  • 71
Truong Hung
  • 21
  • 1
    Are you familiar with the expressions for areas of square and rectangle? – Arpan May 15 '15 at 07:28
  • 2
    Start by expressing the geometric mean of the sides of a rectangle, as well as the sides of square in terms of its (the square's) area. – Ari May 15 '15 at 07:30

2 Answers2

1

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

0

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 $b$?

Jesse P Francis
  • 1,469
  • 3
  • 27
  • 71
  • Hm... then can I say that the side length of a square is length c and so the area of the square is c^2. and let the rectangle side length be a and b so the area for the rectangle is ab. and since they're congruent. c^2 = ab. we take the square root on both side and thus gives us the equal side length that we needed. Is that sufficient? – Truong Hung May 15 '15 at 07:33
  • As you said, taking square root leaves us with $c=\sqrt{ab}$, and $\sqrt{ab}$ is called the geometric mean of $a$ and $b$! – Jesse P Francis May 15 '15 at 07:37