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Example Question #1 : How To Find If Trapezoids Are Similar
Given Trapezoid with bases and . The trapezoid has midsegment , where and are the midpoints of and , respectively.
True, false, or undetermined: Trapezoid Trapezoid .
The fact that the trapezoid is isosceles is actually irrelevant. Since and are the midpoints of legs and , it holds by definition that
It follows that
However, the bases of each trapezoid are noncongruent, by definition, so, in particular,
Assume that is the longer base - this argument works symmetrically if the opposite is true. Let - equivalently, .
Since the bases are of unequal length, . The length of the midsegment is the arithmetic mean of the lengths of these two bases, so
Since , it follows that
This disproves similarity, since one condition is that corresponding sides must be in proportion.