Question

if point e is the midpoint of \\( \overline { a c } \\) and point d is the midpoint of \\( \overline { b c } \\), which expression represents the value of u?
\\( u = \frac { q } { 9 } \\)
\\( u = \frac { 1 } { 2 } s \\)
\\( u = \frac { q } { p } \\)
\\( u = 2 s \\)

Explanation:

Step1: Recall the mid - segment theorem

In a triangle, the mid - segment (a segment connecting the midpoints of two sides of a triangle) is parallel to the third side and half its length.
Here, in \(\triangle ABC\), since \(E\) is the mid - point of \(AC\) and \(D\) is the mid - point of \(BC\), then \(ED\) is a mid - segment of \(\triangle ABC\). And in \(\triangle ABC\), if we consider another pair of mid - points (not relevant here as per the problem of relating \(u\) and \(s\)).
In \(\triangle ABC\), for the line segments \(u\) and \(s\), because of the mid - segment property.

Step2: Apply the mid - segment length relationship

The mid - segment theorem states that if a line segment joins the midpoints of two sides of a triangle, then its length is half the length of the third side. If we assume the relevant triangle relationship (where \(s\) is a mid - segment and \(u\) is related to it), we know that \(u = 2s\) (because the side \(u\) is twice the length of the mid - segment \(s\) that is parallel to it).

Answer:

\(u = 2s\)