Folded Over
A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?
Age
14 to 16
Challenge level
Problem
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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.
Student Solutions
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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. $$\Rightarrow \angle BA^{\prime}C = 45^{\circ} \Rightarrow \angle EA^{\prime}D = 45^{\circ}$$ $$\Rightarrow ED = A^{\prime}D = \sqrt 2 - 1$$