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How big is the yellow triangle?

2021-11-14T13:38:53.125Z


Two routes are drawn in a square. This creates a triangle highlighted in yellow. How big is it in relation to the area of ​​the square?


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Photo: DER SPIEGEL

Last week's physical problem is followed by a not-too-difficult geometrical puzzle.

Two routes are drawn in a square.

One connects a corner point with the center point of another side, the other is an ordinary diagonal - see sketch above.

Both lines and the lower side of the square enclose a triangle that is highlighted in yellow in the drawing.

How big is the area of ​​this triangle in relation to the area of ​​the square?

The ratio is a third.

The solution is quickly found when you realize that the two lines form two triangles that are similar to each other.

We immediately see that the triangle filled in yellow and the filled blue triangle have interior angles of the same size.

Because two angles are formed by lines that intersect two other lines that are parallel to each other - the sides of the square above and below.

See the following drawing:

We know that the top of the blue triangle is half the length of the side of the square, which in turn corresponds to the bottom of the yellow triangle.

Because both triangles are similar, each side of the blue triangle must be exactly half the length of the yellow triangle.

Of course, this also applies to the heights of the two triangles - their ratio is 1: 2.

However, both heights together are as long as one side of the square.

The blue triangle therefore has a height of one third of the side of the square, with the yellow one the length of the height is two thirds.

If we set the side length of the square equal to 1, it also has an area of ​​1. We can calculate the area of ​​the yellow triangle with the formula g * h / 2.

It amounts to:

1 * 2/3 * 1/2 = 1/3.

I discovered this puzzle on the website interestingengineering.com after a hint from the Twitter account Cliff Pickover.

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

  • Which stick is magnetic?

  • Who will get the last coin?

  • Difficult legacy

  • How big is the circular ring?

  • Buying a bicycle with counterfeit money

  • Six acute angles

  • How far is it to the neighboring village?

  • How old is sophie

  • Triangle searched on three circles

  • Pirates fight over 120 gold coins

Source: spiegel

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