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# The hammock

A square inscribed in a triangle. Two areas are given. What is the square’s area?

Scroll down for a solution to this problem.

## Solution

The square area is 2√xy.

This problem can be solved in an algebraic way, as was shown by Dr Rick. The diagram contains three similar triangles: the unshaded ones and the big one. Their hypotenuses have proportions √x : √y : √x+√y. Their areas have proportions as the squares of that and since (√x+√y)2=x+y+2√x√y, the square must have area 2√xy.

Note that this proof also holds in case the square would be parallelogram.

## Poem

The hammock
Keeps our dreams
Keeps our thoughts
From morning
To evening
We lay softly dreaming
Eyes closed, and shivering
Our hands sliding
Our bodies slowly travelling
A bird is kindly singing
A moment of eternity
Far from reality
The hammock is friendly

Bella

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## One reply on “The hammock”

Benjamin Tzursays:

2 (xy)^.5