# Central Median of Trapezoid(Trapezium) and Triangle

**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.

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:**

**Theorem 2:**

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 =>

#### 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.

AA_{1} = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4} = A_{4}A_{5}

We connect A_{5} with B and draw lines through A_{4}, A_{3}, A_{2} and A_{1} that are parallel to A_{5}B. They cross AB respectively in the points B_{4}, B_{3}, B_{2} and B_{1}. These points divide the segment AB into five equal parts. Indeed, from the trapezium BB_{3}A_{3}A_{5} we see that BB_{4} = B_{4}B_{3}. In the same way, from the trapezium B_{4}B_{2}A_{2}A_{4}, we obtain B_{4}B_{3} = B_{3}B_{2}

While from the trapezoid(trapezium) B_{3}B_{1}A_{1}A_{3}, B_{3}B_{2} = B_{2}B_{1}.

Then, from B_{2}AA_{2}, it follows that B_{2}B_{1} = B_{1}A. We finally obtain :

AB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4} = B_{4}B

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.