Folding is fun

Geometry Level 3

A square sheet of paper $ABCD$ is folded such that $B$ falls on the midpoint $M$ of $CD$ . In what ratio does the crease divide $BC$ ?

1:1

2:1

5:3

5:1

1 solution

Brian Charlesworth
Apr 17, 2016

Let the square have side length $x$ . Also let the crease point on side $BC$ be $P$ and $|PB| = y$ .

Then $\Delta PMC$ is a right triangle with $|PM| = |PB| = y, |PC| = x - y$ and $|MC| = \dfrac{x}{2}$ .

By Pythagoras we then have that

$|PM|^{2} = |PC|^{2} + |MC|^{2} \Longrightarrow y^{2} = (x - y)^{2} + \left(\dfrac{x}{2}\right)^{2}$

$\Longrightarrow y^{2} = x^{2} - 2xy + y^{2} + \dfrac{x^{2}}{4} \Longrightarrow 2xy = \dfrac{5x^{2}}{4} \Longrightarrow \dfrac{y}{x} = \dfrac{5}{8} \Longrightarrow \dfrac{x - y}{x} = \dfrac{3}{8}$ .

Thus the desired ratio is $\dfrac{|BP|}{|PC|} = \dfrac{y}{x - y} = \dfrac{5}{3}$ , i.e., $\boxed{5:3}$ .

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For convenience, let the side of the square be 2. From similar triangles BFP and BCM

$\frac{BM}{BC} = \frac{PB}{FB}$ giving $\frac{\sqrt{5}}{2} = \frac{PB}{\sqrt{5}/2}$ giving PB = 5/4 and thus PC = 3/4

Thus the ratio will be 5:3

Ujjwal Rane - 4 years, 11 months ago

Folding is fun

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