# 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.