### MATH 7200: Foundations of Geometry

EXAM #1 - Question #4

**4. Given two points A and
B, construct a square with diagonal AB.**

Click here
for a GSP sketch of the constructed square.

Click here
for a JSP sketch of the constructed square. NOTE: you will need
a java-enabled browser.

Construction:

Given diagonal AB. The lemma below states that
the diagonals of a square are perpendicular bisectors. So, the
first step is to construct the other diagonal by constructing
the perpendicular bisector [see lemma below].

The lemma also states that the diagonals bisect
each other. In other words, the distances from P to each vertex
are congruent. Two vertices of the square are known: points A
and B. So, to construct the other two vertices, we construct a
circle with center at point P with radius of length d(PA). The
last two vertices (points C and D) are found at the intersection
of the circle and the perpendicular bisector.

LEMMA: Given a square ABCD, the diagonals are
perpendicular bisectors of each other. See Proof.

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