Given a trapezoid, a middle line and a diagonal, the diagonal divides the middle line into two parts

Given a trapezoid, a middle line and a diagonal, the diagonal divides the middle line into two parts, one part 6 cm and the second 12 cm, find the base.

The middle line of a trapezoid is a segment whose length is half the sum of its bases:

KM = (AD + BC) / 2.

The diagonal of the trapezoid intersects its midline at point Н and divides this trapezoid into two triangles ∆ABС and ∆AСD.

Thus, the segment KН is the midline of the triangle ∆ABС, НM is the midline of the triangle ∆AСD.

The middle line of the triangle will also connect the midpoints of its lateral sides, equal to half its length. Therefore, we can find the length of their bases:

KN = BC / 2;

BC = KН ∙ 2;

BC = 6 ∙ 2 = 12 cm;

НM = AD / 2;

AD = НM ∙ 2;

BP = 12 ∙ 2 = 24 cm.

Answer: the bases of the trapezoid are 12 cm and 24 cm.