# Central Median of Trapezoid(Trapezium) and Triangle

A quadrilateral with two opposite parallel sides is called a trapezoid(trapezium).

The sides of the trapezoid(trapezium) that are parallel, are called bases and those that are not parallel are called legs. If the legs are equal in length, then this is an isosceles trapezoid.
DE and CF are called heights or altitudes of the trapezoid.

AD and BC are the legs of the trapezoid.
AB and CD are parallel sides.

#### Central Median of Trapezoid(Trapezium)

A line segment joining the middles of the sides of a trapezoid that are not parallel is called a central median.
MN is the central median of ABCD. M is the middle of AB and N is the middle of BC.

AM = MD; BN = NC

MN central median, AB and CD are bases, AD and BC are legs

The central median of a trapezoid is parallel to its sides. In our case - MN || AB || DC.

Theorem 1:

If a line crosses the middle of one of the legs of a trapezoid and is parallel to its bases, then it crosses the middle of the other leg.

Theorem 2:

The central median of a trapezoid(trapezium) is half the lengths of the two parallel sides.

or
$MN = \frac{AB + DC}{2}$

#### Central Median of a Triangle

The line segment joining the middles of two of the sides of a triangle is called central median of a triangle. It is parallel to the third side and its length is half the length of the third side.
Theorem: If a line segment crosses the middle of one side of a triangle and is parallel to another side of the same triangle, then this line segment halves the third side.

AM = MC and BN = NC =>

MN || AB
MN = AB/2

#### Application of the properties of the central medians in a trapezoid(trapezium) and triangle

Dividing a segment into equal parts.

Assignment: Divide the given segment AB into 5 equal parts.
Solution:
Let p be an arbitrary ray with origin A and p does not lie on AB. We draw consecutively five equal segments on p.
AA1 = A1A2 = A2A3 = A3A4 = A4A5
We connect A5 with B and draw lines through A4, A3, A2 and A1 that are parallel to A5B. They cross AB respectively in the points B4, B3, B2 and B1. These points divide the segment AB into five equal parts. Indeed, from the trapezium BB3A3A5 we see that BB4 = B4B3. In the same way, from the trapezium B4B2A2A4, we obtain B4B3 = B3B2

While from the trapezoid(trapezium) B3B1A1A3, B3B2 = B2B1.
Then, from B2AA2, it follows that B2B1 = B1A. We finally obtain :
AB1 = B1B2 = B2B3 = B3B4 = B4B
It is clear that if AB should be divided into another number of equal parts, we should project the same number of equal segments on p. Then we proceed in the way above-described.

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