Question

construct a square withe an area of √18cm2​

Answer 1

Answer:

This problem is known as "Squaring a Rectangle". There are easy mathematical ways to solve the challenge, but the geometric proof this question asks, using only a compass and straightedge, is elegant.

Start with rectangle ABCD.

Extend side AB.

Draw an arc centered at B with radius BC. Let E be the intersection of the arc and ray AB.

Construct the midpoint M of segment AE.

Construct a circle centered at M with radius MA.

Extend side CB.

Let F be the point where the circle and ray CB intersect.

Then BF is the side of the desired square. Complete by constructing a square with side BF.

The Compass Equivalence Theorem states that any geometric construction via a fixed compass can be made with a collapsing compass: Proposition II of Book I of Euclid's Elements.

It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.

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