# Segment MN-middle line of trapezoid ABCD with a large base AD. Find the segment MN

Segment MN-middle line of trapezoid ABCD with a large base AD. Find the segment MN if AB = 3√5 cm, DC = 4√5, and the perimeter of the trapezoid is 21√5

The middle line of a trapezoid is a segment connecting the midpoints of the lateral sides of the trapezoid and located parallel to the bases of the trapezoid.

According to the theorem on the midline of a trapezoid, its length is equal to the half-sum of the lengths of the bases of the trapezoid. MN = 1/2 * (AD + BC).

The perimeter of a trapezoid is the sum of the lengths of its bases and sides. P = AB + BC + CD + DA.

Knowing that the perimeter is 21√5, AB = 3√5, DC = 4√5, we find the sum of the bases AD and BC.

AD + BC = P – (AB + DC);
AD + BC = 21√5 – (3√5 + 4√5);
AD + BC = 21√5 – 7√5;

Find the middle line of the trapezoid.
MN = 1/2 * (AD + BC);
MN = 1/2 * 14√5;
MN = 7√5.

Answer: 7√5 is the length of the segment MN.

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