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Difficulty: MediumProperties of Quadrilaterals

Determine whether the following statement is true or false: In any trapezoid, the line segment connecting the midpoints of the non-parallel sides (the midsegment) is parallel to the bases and has a length equal to half the difference of the lengths of the bases.

Answer: Answer

Answer

The statement is False.
The statement is False because according to the Trapezoid Midsegment Theorem, the midsegment is parallel to the bases and its length is equal to half the sum (the arithmetic mean) of the base lengths, m=b1+b22m = \frac{b_1 + b_2}{2}.

Step-by-Step Solution

1
Identify the geometric shape and segment described in the statement.
The shape is a trapezoid with parallel bases of lengths b1b_1 and b2b_2, and the segment is the midsegment connecting the midpoints of the non-parallel legs.
Understanding the definition of a trapezoid's midsegment is necessary to evaluate its properties.
2
State the Trapezoid Midsegment Theorem.
The Trapezoid Midsegment Theorem states that the midsegment is parallel to both bases and its length mm is equal to the average of the two base lengths: m=b1+b22m = \frac{b_1 + b_2}{2}.
This theorem directly provides the exact mathematical relationship for the length of the midsegment.
3
Compare the theorem with the statement provided in the stem.
The statement claims the length is half the difference, ∣b1−b2∣2\frac{|b_1 - b_2|}{2}, which contradicts the actual formula involving the sum.
Since the statement uses subtraction instead of addition, the statement is false.

Key Concept

Trapezoid Midsegment Theorem
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Topics

Properties of Quadrilaterals

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