Since the shape and size of a plane figure is invariate under coordinate translations and rotations, a general trapezoid can be placed with one vertex at the origin and one base coincident with the -axis without loss of generality. See figure 1: Figure 1Using the Midpoint formulae, the coordinates of the endpoints of the median are established as shown in figure 2: Figure 2:
Since the line segments forming the bases and the median are horizontal lines, the measures can be determined by simple differences of the -coordinates. The measure of the lower base is simply , the measure of the upper base is . Half of the sum of the bases is then . Compare with the measure of the median: Q.E.D. John
Let #ABCD# be a trapezoid with lower base #AD# and upper base #BC#. Connect vertex #B# with midpoint #N# of opposite leg #CD# and extend it beyond point #N# to intersect with continuation of lower base #AD# at point #X#. Consider two triangles #Delta BCN# and #Delta NDX#. They are congruent by angle-side-angle theorem because Therefore, segments #BC# and #DX# are congruent, as well as segments #BN# and #NX#, which implies that #N# is a midpoint of segment #BX#. But #AX# is a sum of lower base #AD# and segment #DX#, which is congruent to upper base #BC#. Therefore, #MN# is equal to half of sum of two bases #AD# and #BC#. End of proof. The lecture dedicated to this and other properties of quadrilaterals as well as many other topics are addressed by a course of advanced math for high school students at Unizor.
Answer: B. The midsegment theorem Step-by-step explanation: In a trapezoid, the midline is parallel to the bases and its length is half their sum. Hope it helps |