Mission (Im)possible

Geometry Level 2

Is it possible to draw these figures?

I. A triangle of side ratio 1 : 2 : 3 \text{ I. A triangle of side ratio 1 : 2 : 3 }

II. A quadrilateral of side ratio 1 : 2 : 3 : 4 \text{ II. A quadrilateral of side ratio 1 : 2 : 3 : 4 }

Both I. & II. are impossible. I. is possible, but II. is impossible. I. is impossible, but II. is possible. Both I. & II. are possible.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

In order to draw a triangle, the length sum of two lesser sides must be more than the longest one. In the first scenario, 1+2 is only equal to 3, so the best we can make is a line:

In order to draw a rectangle, on the other hand, if we put side lengths 1 (BC) & 4 (AD) across each other, we can draw a quadrilateral as shown below:

Hence, case I. is impossible, but case II. is possible.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

An alternative approach is to show that there exists a cyclic quadrilateral with sides 1, 2, 3 and 4. This is immediate by Brahmagupta's fomula .

Pi Han Goh - 5 years, 1 month ago

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