Square Slicing

Geometry Level 1

Consider the figure on the right. The goal is to cut off a piece and rearrange it into a square.

Clearly, we can cut off the triangle with interior angles 30 $^\circ$ -45 $^\circ$ , and paste on the at the bottom to form a square, as shown on the left.

Is it possible to cut of a different piece from the original shape and rearrange it into a square?

No, it is not possible Yes, it is possible

1 solution

Michael Huang
Jan 27, 2018

The cut is colored orange in this diagram.

Since the "removed" triangle can be rotated $45^{\circ}$ about the lower right vertex, then we can "fit" the new triangle between the replaced and the removed. So since all these triangles share the common areas, angles and side lengths, we can use the combination of two triangles to assemble back into the square.

This solution doesn't seem to fit the guidelines. My opinion.

ALBERT BOYLES - 3 years, 4 months ago

I think I am seeing how your solution fits but can you provide the 'new' square formed from these pieces that you cut?

A Former Brilliant Member - 3 years, 4 months ago

You are right !

Ayman Sossi - 3 years, 4 months ago

I need more pictures. Proposed solution doesn't convince me.

Ron Berry - 3 years, 4 months ago

Which part doesn't seem to be clear to you? Are you assuming that the solution isn't enough?

Michael Huang - 3 years, 4 months ago

Show me a picture with the new cut line and how the two pieces fit together.

Ron Berry - 3 years, 4 months ago

if we number the triangles 1,2,3 starting from bottom, being 1 empty, and triangles 2 and 3 forming 1 piece, then if we rotate this piece, 2 fits in 1, and 3 fits in 2, cause all triangles are congruent (the same). This works cause the 45 º angle, and the original shape being a square (90º angles)

Eliud Alejandro Maldonado Sanchez - 3 years, 4 months ago

Square Slicing

1 solution

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