In trapezoid ABCD, the base AD = 10, the middle line EF intersects with the diagonal BD at point O

In trapezoid ABCD, the base AD = 10, the middle line EF intersects with the diagonal BD at point O, the difference between the segments EO and OF is 3. Find the middle line EF

Consider a trapezoid ABCD.

According to the problem statement, EF is the middle line of the trapezoid and

AD = 10, EO – OF = 3.

By the property of the middle line of the trapezoid, we have that the straight line EF is parallel to the bases of the trapezoid. Hence it follows that

EO is the middle line of triangle ABD.

By the property of the middle line of the triangle, we have that

EO = 1/2 * AD = 1/2 * 10 = 5.

By the condition of the problem:

EO – OF = 3,

5 – OF = 3,

OF = 2.

But the middle line of the trapezoid ABCD:

EF = EO + OF = 5 + 2 = 7.

Answer: EF = 7.