# Construction of the Root-Mean-Square in a Trapezoid

## by Shannon Umberger

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

Construct a line perpendicular to base MN at
point M. Construct a circle with center at point M and with radius
= a. Construct the intersection of this circle with the previous
line and call it point R. Construct segment RN. By the Pythagorean
Theorem, RN = sqrt(a^2 + b^2).

Construct the midpoint of segment RN and call
it point S. Construct a line perpendicular to segment RN at point
S. Construct a circle with center at point S and passing through
point N. Construct the intersection of this circle and the perpendicular
line and call it point T. Construct segment NT. By construction,
SN = ST = (sqrt(a^2 + b^2))/ 2. By the Pythagorean Theorem, NT
= sqrt((a^2 + b^2)/ 2).

Construct a circle with center at point N and
passing through point T. Construct the intersection of this circle
and base NM and call it point U. Since segments NT and NU are
radii of the same circle, then NT = NU = sqrt((a^2 + b^2)/ 2).

Construct a line parallel to leg KN through
point U. Construct the intersection of this line and leg LM and
call it point Q. Construct a line parallel to base NM through
point Q. Construct the intersection of this line and leg KN and
call it point P. By construction, quadrilateral PQUN is a parallelogram,
and so PQ = NU = sqrt((a^2 + b^2)/ 2).

Construct segment PQ. The length of this segment
is the root-mean-square, "r," of the bases KL and NM.

Double check the construction by taking measurements
and using the equation for finding the root-mean-square.

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