Intermediate Geometry : How to find if trapezoids are similar

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #276 : Intermediate Geometry

Given Trapezoid  with bases  and . The trapezoid has midsegment , where  and  are the midpoints of  and , respectively.

True, false, or undetermined: Trapezoid  Trapezoid .

Possible Answers:

Undetermined

True

False

Correct answer:

False

Explanation:

The fact that the trapezoid is isosceles is actually irrelevant. Since  and  are the midpoints of legs  and , it holds by definition that 

and 

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

,

and

.

This disproves similarity, since one condition is that corresponding sides must be in proportion.

 

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