# A straight line is drawn through the apex M of the CMR trapezium (larger base-CT), parallel to the lateral side

**A straight line is drawn through the apex M of the CMP trapezium (larger base-CT), parallel to the lateral side of the PT. This line and the base of the CT intersect at point E. The perimeter of the triangle KME is 17 cm, MP = 7 cm, KE = 4 cm. Calculate: a) the length of the midline of the trapezoid; b) the perimeter of the trapezoid**

In the quadrangle EMPT MP is parallel to ET, since ET lies on the lower base of the CT trapezoid, and ME is parallel to PT by construction, then EMPT is a parallelogram, which means ME = PT, ET = MP = 7 cm.

Base KP = KE + ET = 4 + 7 = 11 cm.

Let’s define the middle line of the trapezoid.

AB = (MR + CT) / 2 = (7 + 11) / 2 = 9 cm.

The perimeter of the trapezoid is: P = (KM + MR + RT + CT).

Since PT = ME, and KT = KE + ET, then P = (KM + ME + KE + MP + ET) = Pkm + 2 * MP = 17 + 14 = 31 cm.

Answer: The middle line of the trapezoid is 9 cm, the perimeter of the trapezoid is 31 cm.