compass and straightedge construction of square
One can construct a square with sides of a given length s using compass and straightedge as follows:
1.Draw a line segment of length s. Label its endpoints P and Q.
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P
Q
2.Extend the line segment past Q.
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P
Q
3.Erect the perpendicular to PQ−→− at Q.
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P
Q
4.Using the line drawn in the previous step, mark off a line segment of length s such that one of its endpoints is Q. Label the other endpoint as R.
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P
Q
R
5.Draw an arc of the circle with center P and radius PQ�����.
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P
Q
R
6.Draw an arc of the circle with center R and radius QR����� to find the point S where it intersects the arc from the previous step such that S≠Q.
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P
Q
R
S
7.Draw the square PQRS.
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P
Q
R
S
This construction is justified because PS=PQ=QR=QS, yielding that PQRS is a rhombus. Since ∠PQR is a right angle, it follows that PQRS is a square.
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