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Quite weird - puzzle of the week

2021-08-15T07:43:49.369Z


The diameter of a circle divides a chord into two parts. These sections x and y have an amazing property. Why?


Enlarge image

Photo: DER SPIEGEL

In math, crazy things happen sometimes.

So today: You draw the diameter in a circle with radius 1.

And a second chord that cuts the diameter at an angle of 45 degrees - see drawing above.

The diameter divides the chord into two parts with lengths x and y.

But no matter where exactly the chord and diameter meet: as long as the angle between them is 45 degrees, the

sum x2 + y2

does not change.

Why not?

How big is this sum?

The sum

x2 + y2

is 2. Always.

One possible solution uses the mid-perpendicular of the tendon AB.

It divides this into two pieces of equal length with the length (x + y) / 2 also runs through the center of the circle M - see the following drawing:

Two right-angled triangles have been created through the vertical center line: CDM and ACM.

The CDM triangle is even isosceles because the angles at points D and M are both 45 degrees.

The track CD has the length

(x + y) / 2 - y = (xy) / 2

This then also corresponds to the length of the line CM in the right-angled triangle ACM, because CD = CM.

With this we also know the lengths of all three sides of the right triangle ACM.

According to the Pythagorean theorem:

AC2 + CM2 = 12

We now substitute (x + y) / 2 for AC and (xy) / 2 for CM and get:

(x2 + 2xy + y2 + x2 - 2xy + y2) / 4 = 1

We multiply both sides by 4. The terms + 2xy and - 2xy cancel each other out:

2x2 + 2y2 = 4


x2 + y2 = 2

I discovered this puzzle in the Facebook group "Math problems and puzzles" of the Russian puzzle expert Konstantin Knop.

If you missed a puzzle from the past few weeks, here are the most recent episodes:

  • The perfect egg

  • Cleverly asked

  • Enclosed by six circles

  • x, y unsolved

  • Floor plan with gaps

  • Wrong track

  • Seven in one go

  • Hundred square numbers

  • On the brink

  • Secret access code

  • Who lies?

    Who is telling the truth

Source: spiegel

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