The midline of a triangle divides it into two parts — a triangle and a trapezoid. This trapezoid is
also divided into two parts by its midline. As a result we obtain three parts — one triangle and two
trapezoids. The areas of two of these parts are integers. Prove that the area of the third part is also
an integer.
Answers
Step-by-step explanation:
Midline and Midsegment of Trapezoid and Triangle
A quadrilateral with two opposite parallel sides is called a trapezoid(trapezium).
Trapezoid
The parallel sides of the trapezoid, are called bases (AB and CD) and the ones that are not parallel are called legs (AD and BC).
If the legs are equal in length, the trapezoid is called isosceles.
DE and CF are altitudes.Midline of Trapezoid
Midline of trapezoid (trapezium)
A line that joins the midpoints of the sides that are not parallel is called a midline(or a midsegment) of trapezoid.
The line MN is the midline of ABCD. And the segment MN is the midsegment of ABCD.
AM = MD
BN = NC
The midline of a trapezoid is parallel to its sides.
In our case - MN || AB || DC.
Theorem 1:
If a line through the midpoint of a leg of a trapezoid is parallel to its bases, then the line passes through the midpoint of the other leg.
Theorem 2:
The midsegment of a trapezoid is half the lengths of the two parallel sides.
In other words:
\displaystyle \overline{MN} = \frac{\overline{AB} + \overline{DC}}{2}
MN
=
2
AB
+
DC