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Photo: DER SPIEGEL
Two squares of the same size - that's all we need for this week's pretty puzzles.
It's not too difficult if you haven't completely forgotten the basics of geometry.
The two squares are exactly on top of each other - then one is turned to the left over the lower left corner.
The upper left corner of the unrotated square is then inside the rotated square.
This corner forms an angle together with two opposite points of the rotated square.
In the drawing above, this angle is highlighted in yellow.
How big is this angle?
Does its size depend on how far that one square was turned to the left?
The angle has a size of
135 degrees
.
The size of the angle does not change as long as the vertex of the yellow marked angle lies within the tilted square.
Because we are turning a square around its lower left corner to the left, the three points that form the yellow angle lie on a semicircle - see the following drawing.
They form what is known as a tendon triangle.
The yellow angle you are looking for is above the chord, which is a diagonal of the tilted square.
Mathematicians call the yellow angle circumferential or peripheral angle.
According to the set of circumferential angles, all circumferential angles over an arc are the same.
It is therefore already clear that the yellow angle does not depend on the specific tilt angle of the square as long as the rotation is greater than 0 degrees and less than 90 degrees.
But how do we calculate the size of the angle?
The two legs of the yellow angle and the two lower sides of the tilted square form a square with a right angle.
This is marked in white in the drawing.
This square is composed of two isosceles triangles, the base angles of which are each the same size.
In the drawing above, these angles are marked in red and blue.
The yellow angle you are looking for corresponds to the sum of the red and blue angles.
The inside angle sum in the rectangle is 360 degrees.
So:
360 = 90 + red + red + blue + blue
270 = 2 * (red + blue)
135 = red + blue = yellow
I discovered the geometry puzzle on Twitter.
It goes back to the math teacher and geometry puzzle collector Catriona Agg.
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Who is telling the truth
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