Theorem 6. The median of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases. Example: Given: Trapezoid MINS with median
Prove:
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The Correct Answer and Explanation is:
Correct Answer:
Given: Trapezoid MINSMINSMINS with median TR‾\overline{TR}TR
To Prove:
- TR‾∥IN‾\overline{TR} \parallel \overline{IN}TR∥IN
- TR‾∥MS‾\overline{TR} \parallel \overline{MS}TR∥MS
- TP=12(MS‾+IN‾)TP = \frac{1}{2} (\overline{MS} + \overline{IN})TP=21(MS+IN)
Explanation
Theorem 6 in geometry states that the median of a trapezoid is parallel to both bases, and its length is the average (one-half the sum) of the lengths of the two bases.
In the given figure, MINSMINSMINS is a trapezoid where MS‾\overline{MS}MS and IN‾\overline{IN}IN are the two bases, and TR‾\overline{TR}TR is the median. The median connects the midpoints of the non-parallel sides MI‾\overline{MI}MI and SN‾\overline{SN}SN. This is evident from point TTT lying midway on MI‾\overline{MI}MI and point RRR lying midway on SN‾\overline{SN}SN.
By Theorem 6, the median TR‾\overline{TR}TR must be parallel to both IN‾\overline{IN}IN and MS‾\overline{MS}MS. Thus, we have:
- TR‾∥IN‾\overline{TR} \parallel \overline{IN}TR∥IN
- TR‾∥MS‾\overline{TR} \parallel \overline{MS}TR∥MS
This parallelism arises from the properties of similar triangles and midsegment theorems. Since TTT and RRR are midpoints, the line segment connecting them (i.e., TR‾\overline{TR}TR) must be parallel to both bases.
Furthermore, the length of the median TR‾\overline{TR}TR is the average of the lengths of the two bases. This leads to the equation:TP=12(MS‾+IN‾)TP = \frac{1}{2} (\overline{MS} + \overline{IN})TP=21(MS+IN)
This is a direct application of the trapezoid midsegment theorem, which ensures that the segment connecting the midpoints of the non-parallel sides of a trapezoid has a length equal to the average of the two bases.
In summary, the problem illustrates a fundamental property of trapezoids: the median is both parallel to the bases and has a length equal to the mean of the bases.
