The larger base of the trapezoid is 14 cm long. Find the length of the smaller base

The larger base of the trapezoid is 14 cm long. Find the length of the smaller base if the distance between the midpoints of the diagonals of the trapezoid is 3 cm.

Let’s draw the middle line of the HP trapezoid ABCD. The middle line of the trapezoid divides the diagonals in half, then, by condition, KM = 3 cm.

The segments НК and РМ are the middle lines of the triangles ABC and BCD, and then НК = РМ = BC / 2.

Let the length of НK = РM = X cm, then BC = 2 * X cm, and HP = (X + 3 + X) = (2 * X + 3) cm.

According to the formula of the middle line of the trapezoid: НР = (ВС + АD) / 2.

2 * X + 3 = (2 * X + 14) / 2.

4 * X + 6 = 2 * X + 14.

2 * X = 8.

X = НK = PM = 8/2 = 4 cm.

Then BC = 4 * 2 = 8 cm.

Answer: The length of the smaller base is 8 cm.