12
The midline of a trapezoid
Textbook: pp. 34–35
GoalKnow the midline theorem for a trapezoid and use it in problems.
New words
midline of a trapezoid · trapetsiyaning o‘rta chizig‘isum of the bases · asoslar yig‘indisimidpoint of a leg · yon tomon o‘rtasimidpoints of the diagonals · diagonallar o‘rtalari
Explanation
The segment joining the midpoints of the legs of a trapezoid is called its midline. Theorem: the midline of a trapezoid is parallel to the bases and equals half the sum of the bases, that is m = (a + b) : 2. In the proof, a line through a vertex of the smaller base meets the extension of the larger base, and the midline theorem for a triangle is used. Consequence: a line through the midpoint of one leg parallel to the bases also bisects the other leg. In addition, the segment joining the midpoints of the diagonals is parallel to the bases and equals half their difference, (a − b) : 2 with a > b.
Worked examples
The bases of a trapezoid are 13 cm and 27 cm. Its midline is (13 + 27) : 2 = 20 cm.
The midline of a trapezoid is 18 cm and one base is 14 cm. The bases add up to 2 · 18 = 36 cm, so the other base is 36 − 14 = 22 cm.
Class activity
“Find the middle”: fold the legs of a paper trapezoid to find their midpoints, join them and compare the midline with the bases.
Practice
1
The bases of a trapezoid are 11 cm and 19 cm. How long is the midline (cm)?
15
2
The midline of a trapezoid is 21 cm and one base is 16 cm. How long is the other base (cm)?
26
3
The bases of a trapezoid are 40 cm and 18 cm. How long is the segment joining the midpoints of the diagonals (cm)?
11
4
Why is the midline longer than the smaller base and shorter than the larger?
The average of two numbers lies between them: (a + b) : 2 is larger than the smaller and smaller than the larger.