# Sunday Times Teaser 3002 – Short Cut

### by Victor Bryant

#### Published Sunday April 05 2020 (link)

To demonstrate a bit of geometry and trigonometry to my grandson, I took a rectangular piece of paper whose shorter sides were 24 cm in length. With one straight fold I brought one corner of the rectangle to the midpoint of the opposite longer side. Then I cut the paper along the fold, creating a triangle and another piece. I then demonstrated to my grandson that this other piece had double the area of the triangle.

How long was the cut?

1.  The diagram above shows the folded corner and illustrates that the fold produces two identical triangles and a set of angle relationships that allow the teaser to be solved. Using the symbols introduced in the diagram, we can see that:

$y=s\sin{2\alpha}$$x=2t\sin{2\alpha}$

Combining these two equations we obtain:

$x\,y = 2\,s\,t\,\sin^2{2\alpha}$

With $$A_r$$ and $$A_t$$ as the areas of the rectangle and the cut off triangle respectively, this equation becomes$A_r = 4A_t\,\sin^2{2\alpha}$showing that:

$\alpha=\frac{1}{2}\sin^{-1}{ \left( \frac{\sqrt{A_r/A_t}}{2} \right)}$

From the teaser text we know that $$A_r-A_t=2A_t$$, giving $$A_r/A_t = 3$$ and showing from this equation that $$\alpha=30^o$$.

From the diagram we can see that$\begin{array}{l}y&=&s\,\sin{2\,\alpha}\\&=&h\,\cos{\alpha}\sin{2\,\alpha}\\&=&2\,h\,\sin{\alpha}\cos^2{\alpha}\\ \end{array}$ With $$\alpha=30^0$$ this shows that the length of the cut line $$h$$ is equal to $$4\,y/3$$, which gives the teaser solution as 32cm.

Other lengths are given by:$t\,=\,2\,y/3$$s\,=\,x\,=\,2\,y/\sqrt{3}$ with lengths of 16cm and ~27.71cm respectively.

2. From the areas, 3 s (t / 2) = x y
From similar triangles s / y = t / (x / 2)
Eliminating s and y leads to t = x / sqrt(3)
And so alpha = 30 degrees and all of the triangles are similar.

From the bottom left triangle, t / (y – t) = 2 and so t = 2 y / 3
The cut = h = 2 t = 4 y / 3 = 32 cm

3. 