A Suggestion

Take a segment of lengtha + bas the diameter of a semicircle. Construct a perpendicular from the common end of the segments of lengthaand lengthband locate its intersection with the semicircle. Draw the segment from the midpoint of the diameter to that intersection.

The blue segment is always as long or longer than the red segment, equality if and only if

a = b.

Click

herefor a GSP file to vary the lengths ofaandb.

Construct your own GSP File and animation.

EXTENSION:Consider adding an altitude to the right triangle as shown:

1. Find

hin terms ofaandb.2. Show h is less than or equal to the geometric mean (the red segment) with equality if and only if

a=b.Click

herefor a GSP file.Construct your own GSP file.

EXTENSION:

Given a rectangle with lengthaand widthb, construct a square of the same area.