# Construction of the Heronian Mean in a Trapezoid

## by Shannon Umberger

Given trapezoid KLMN, with KL // NM, KL = a,
and NM = b.

Construct the arithmetic mean segment (in red)
and the geometric mean segment (in yellow).

Construct the mipoint of the arithmetic mean
segment and call it point R. Construct a line perpendicular to
the arithmetic mean segment at point R. Construct the intersection
of this line and the geometric mean segment and call it point
S.

Construct segment RS. Construct the midpoint
of this segment and call it point T.

Construct a circle with center at point R and
passing through point T. Construct the intersection of this circle
and the arithmetic mean segment and call it point U. Construct
a circle with center at point U and passing through point R. Construct
the intersection of this circle and the arithmetic mean segment
and call it point V. Constrct a circle with center at point V
and passing through point U. Construct the intersection of this
circle and the arithmetic mean segment and call it point W. Since
segments RU, UV, and VW are all radii of congruent circles by
construction, then RU = UV = VW = (1/ 3)RW.

Construct segment SW. Construct a line parallel
to SW through point U. Construct the intersection of this line
and segment RS and call it point X. Since RU = (1/ 3)RW and triangles
XRU and SRW are similar by construction, then RX = (1/3)RS.

Construct a line parallel to base NM through
point X. Construct the intersection of this line and leg KN and
call it point P. Construct the intersection of this same line
and leg LM and call it point Q.

Construct segment PQ. The length of this segment
is the Heronian mean, "h," of the bases KL and NM.

Double check the construction by taking measurements
and using the equation for finding the Heronian mean.

**Return
to Essay # 3 - Some "Mean" Trapezoids**