Question

quadrilateral abcd is an isosceles trapezoid with median ef. which of the following statements is true?
select the best answer from the choices provided.
a. ef is parallel to the bases.
b. ef = 2(ab + dc)
c. \\( \frac { 1 } { 2 } e f = a b + d c \\)
d. \\( e f = \frac { 1 } { 2 } ( a d + c b ) \\)

Explanation:

Step1: Recall the properties of the median of a trapezoid

The median (or mid - segment) of a trapezoid has two important properties:

1. It is parallel to the two bases.
2. Its length is equal to the average of the lengths of the two bases, i.e., \(EF=\frac{1}{2}(AB + DC)\)

Step2: Analyze each option

• Option A:

By the property of the median of a trapezoid, the median \(EF\) is parallel to the bases \(AB\) and \(DC\).

• Option B:

Since \(EF=\frac{1}{2}(AB + DC)\), then \(2EF=AB + DC\), not \(EF = 2(AB+DC)\)

• Option C:

Since \(EF=\frac{1}{2}(AB + DC)\), then \(2EF=AB + DC\), not \(\frac{1}{2}EF=AB + DC\)

• Option D:

The formula for the length of the median of a trapezoid is related to the lengths of the bases (\(AB\) and \(DC\)), not the non - parallel sides (\(AD\) and \(BC\)). The correct formula is \(EF=\frac{1}{2}(AB + DC)\)

Answer:

A. \(EF\) is parallel to the bases.