Folded Over
A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?
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![Folded Over Folded Over](/sites/default/files/styles/large/public/thumbnails/content-id-2437-folded%252520over%252520Stefania.png?itok=_Ks4HbUj)
A rectangular sheet of paper with sides $1$ and $ \sqrt{2} $ has been folded once as shown, so that one corner just meets the opposite long edge. What is the value of the length $d$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
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![Folded Over Folded Over](/sites/default/files/styles/large/public/thumbnails/content-id-2437-folded%252520over%252520sol%252520Stefania.png?itok=gI0Mlz7K)
The fold is made along $BE$. $A$ folds onto $A^{\prime}$.
$$A^{\prime} B = AB = \sqrt 2 \Rightarrow A^{\prime}C = 1$$ (by Pythagoras)
$$\Delta A^{\prime}BC$$ is therefore a right-angled isosceles triangle. $$\Rightarrow \angle BA^{\prime}C = 45^{\circ} \Rightarrow \angle EA^{\prime}D = 45^{\circ}$$ $$\Rightarrow ED = A^{\prime}D = \sqrt 2 - 1$$