Last modified on June 11th, 2022

chapter outline

 

Midsegment (Median) of a Trapezoid

The mid-segment of a trapezoid is the line segment mid-way between its two bases.  It is also called the median or the midline.

Midsegment or Median of a Trapezoid

How many mid-segments does a trapezoid have?

A trapezoid has only one mid-segment since it has only 2 parallel sides and 2 non parallel sides.

The mid-segment bisects the non parallel sides. In other words, a mid-segment is the mid-way between the two bases.

The next segment will help you find the mid-segment of a trapezoid.

Formula

The basic formula to find the mid-segment of a trapezoid is given below:

How to Find the Midsegment of a Trapezoid

Find the length of the mid-segment of the trapezoid given.

Solution:

As we know,
Mid-segment (M) = ½(a + b), here a = 23 cm, b = 14 cm
∴ M = ½(23 + 14)
= 18.5 cm

In trapezoid PQRS, what is the length of mid-segment XY.

Solution:

As we know,
Mid-segment (M) = ½(27 + 12), here a = 27 cm, b = 12 cm
∴ M = ½(27 + 12)
= 19.5 cm

Trapezoid Midsegment Theorem

The trapezoid mid-segment theorem states that a line connecting the midpoints of the non-parallel sides (legs) is parallel to the bases. It measures half the sum of lengths of the bases.

Trapezoid Midsegment Theorem Proof

Prove that the line connecting the midpoints of the legs of a trapezoid ABCD is parallel to the bases and half the sum of lengths of the bases.

To prove:

EF ∥AD, EF ∥BC

Proof:

AB ∥ CD, AE  ≅ EB, DF ≅ FC
AF and BC produced to point P

StatementReason
∠ADF ≅ ∠FCPAAS (Angle-Angle-Side) postulate, alternate angles
∠AFD ≅ ∠CFPAAS (Angle-Angle-Side) postulate, opposite angles
ΔADF ≅ ΔFCPAAS postulate
EF ∥BP ∥BC EF = ½ BPEF is mid-segment of ΔABP
EF = ½(BC + CP)CP ≅ AD, SS postulate, ΔADF ≅ ΔFCP
EF = ½(AD + BC)  AD ≅ CP Hence proved
EF ∥AD, EF ∥BC  AD ∥BC, given Hence proved

Last modified on June 11th, 2022

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