If line n bisects \(\overline { CE} \) , find CD.

1. Since line \(n\) bisects \(\overline { CE} \), \(CD\) is equal to \(DE\).
2. Set the expressions for \(CD\) and \(DE\) equal to each other:
   \[x + 6 = 4x - 21\]
3. Solve for \(x\):
   \[x + 6 = 4x - 21\]
   \[6 + 21 = 4x - x\]
   \[27 = 3x\]
   \[x = 9\]
4. Substitute \(x\) back into the expression for \(CD\):
   \[CD = x + 6 = 9 + 6 = 15\]

So, \(CD\) is 15.

Supplemental Knowledge

In geometry, a bisector is a line that divides a segment into two equal parts. When a line bisects a segment, it means that the two resulting segments are congruent (equal in length).
Given that line \(n\) bisects \(\overline { CE} \), it means that \(CD = DE\). This equality can be used to set up an equation to solve for the variable involved.

Real-World Applications 

Imagine designing a garden and wanting to place a fountain exactly in the center of a rectangular plot. In order to ensure perfect placement, measure one side of the rectangle before finding its midpoint; placing your fountain here ensures symmetry and balance within your design.
Understanding bisectors can also prove valuable for other fields like architecture and engineering as well as in daily tasks, like evenly dividing objects or spaces.

Understanding geometric concepts such as bisectors is vital for solving real-world issues efficiently, so if you want to broaden your knowledge base in geometry or any other subject area, getting acquainted with bisectors could help immensely.
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