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Intermediate

The divided diagonal

A square diagonal is divided in three by a 45-degree angle. Two pieces are given, what is the length of the third?

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Solution

The third part has length 4.

The trick is to rotate triangle DKC to the other side as shown. Now drawing the segment IJ, one can deduce from SAS that triangles KIC and JIC are congruent.

This gives us the hypotenuse of right triangle IBJ. Finally, the Pythagorean theorem produces x=4.

Visual solution

Geometrical artist David Andriana kindly provided the following visual solution to The divided diagonal:

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