Question

e is the midpoint of \\(\overline{gk}\\) and i is the midpoint of \\(\overline{hj}\\). what is the length of \\(\overline{ek}\\)?
(there is a circle with center f. points h, j, k, g are on the circle. segment hj has midpoint i, with hi = 7 and i is the midpoint. segment gk has midpoint e. the segments fi and fe are from center f to midpoints i and e respectively.)
\\(ek = \square\\)

Explanation:

Step1: Recall the property of chords and mid - points in a circle

In a circle, if a line from the center is perpendicular to a chord, it bisects the chord. Also, chords that are equidistant from the center are equal in length. Here, since \(E\) is the mid - point of \(GK\) and \(I\) is the mid - point of \(HJ\), and the line from the center \(F\) to the mid - point of a chord is perpendicular to the chord. Chords \(GK\) and \(HJ\) are equidistant from the center (the distance from the center to the chord is the length of the line segment from the center to the mid - point of the chord).

Step2: Use the property of equal chords

Chords that are equidistant from the center of a circle are equal in length. The length of \(HJ\) is given as \(2\times7 = 14\) (because \(I\) is the mid - point of \(HJ\), so \(HJ=2\times HI\)). Since \(GK = HJ\) (chords equidistant from the center are equal) and \(E\) is the mid - point of \(GK\), then \(EK=\frac{1}{2}GK\).
Since \(GK = HJ\) and \(HJ = 14\), then \(EK=\frac{1}{2}\times14\)

Answer:

\(7\)