## Teaser 3002: Short-cut

**From The Sunday Times, 5th April 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?

[teaser3002]

## Jim Randell 5:40 pm

on3 April 2020 Permalink |If we assume the fold goes from one of the corners of the rectangle, then we get a nice answer. (See:

Enigma 1402). I don’t think other constructions are possible. [This assumption is justified below].Brainteaser 1798is a puzzle with a similar theme.The triangles

X, Y, Zare all right-angled. We need to find the value ofh, the hypotenuse of triangleX.The area of triangle

Xmust be the same as the sum of the areas of trianglesYandZ, so:From triangle

Z:(So the long side of the rectangle is

2b = 16√3, approximately 27.71 cm).And from triangle

X:Solution:The length of the cut is 32 cm.And we can then demonstrate that

X=Y+Zby constructingXfromYandZ:X,Y,Zare all (30°, 60°, 90°) triangles. The same shape as one of the standard set squares.LikeLike

## Jim Randell 3:39 pm

on5 April 2020 Permalink |Adding a 24-by-2

xstrip on the left-hand side of the diagram (to account for the fold not going from a corner), and adjusting the bases of trianglesYandZto(b – x)and(b + x)leads to the following 2 equations:These can only be satisfied (for positive

a, b) ifx = 0anda = 8(i.e.ais 1/3 the height of the rectangle), as above.So the fold must go from one of the corners, and the assumption above is justified.

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## Benet Allen 7:49 pm

on5 April 2020 Permalink |There’s only one shape where you can fold a corner to the midpoint of the opposite side and be left with a triangle. That’s a 2×1 rectangle. And surely, the remaining piece will have three times the area of the triangle? Am befuddled.

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## Jim Randell 9:23 pm

on5 April 2020 Permalink |My diagram above [ link ] is actually to scale. If you print it out and cut out the rectangle you will find that you can fold the white

Xonto the greyX, and then foldYandZon top (or behind) to make another copy ofX, neatly demonstrating thatX=Y+Zas required.LikeLike