Question

chords and arcs

find the length of \\(\overline{cd}\\).

\\(cd = ?\\)

Explanation:

Identify the properties of the perpendicular segments from the center

The diagram shows a circle with two chords, \(AB\) and \(CD\).
Segments are drawn from the center of the circle perpendicular to both chords.
The length of both perpendicular segments is \(6\).
Since the perpendicular distances from the center to the chords are equal, the chords are equidistant from the center.

Apply the Equidistant Chords Theorem

According to the Equidistant Chords Theorem, in the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Since both perpendicular distances are \(6\), we have:

$$ AB = CD $$

Find the length of chord AB

A line segment from the center of a circle perpendicular to a chord bisects the chord.
The perpendicular segment meets chord \(AB\) at point \(E\), dividing it into two equal segments: \(AE\) and \(EB\).
Given that \(EB = 5\), we have:

$$ AE = EB = 5 $$

Therefore, the total length of chord \(AB\) is:

$$ AB = AE + EB = 5 + 5 = 10 $$

Determine the length of chord CD

Since the chords are equidistant from the center, their lengths are equal:

$$ CD = AB = 10 $$

Answer:

Find the length of \(\overline{CD}\).

\(CD =\) <blank>10</blank>