To find the length of \( DC \) in the parallelogram (which is a rhombus in this case, since the diagonals of a rhombus bisect each other at right angles and all sides are equal), we can utilize the given lengths:
- \( AD = 13 \)
- \( DB = 24 \)
Since \( AD \) and \( BC \) are opposite sides of the parallelogram (rhombus) and are equal, we also have \( BC = 13 \).
To apply the properties of a parallelogram, specifically a rhombus, we can focus on the diagonals \( BD \) and \( AC \). In a rhombus, the diagonals bisect each other, meaning that each side of the diagonal is half of its original length.
Given the length of diagonal \( BD = DB = 24 \), the segments \( DB \) and \( BD \) are equal, and both are divided equally by the intersection point of the diagonals.
Thus, we find: \[ DB = \frac{24}{2} = 12 \] This means that the segment \( DB \) (which corresponds to the intersection point of the diagonals) creates two equal segments: \( DB = 12 \) and \( BD = 12 \).
To find \( DC \) through the use of triangle properties (specifically Triangle \( BDC \)), we apply the property of triangles in which \( BC = 13 \) and we analyze Triangle \( BDC \). We notice that: \[ BC^2 = BD^2 + DC^2 \] Where \( BC = 13 \) and \( BD = 12 \): \[ 13^2 = 12^2 + DC^2 \]
Calculating this gives: \[ 169 = 144 + DC^2 \] \[ DC^2 = 169 - 144 \] \[ DC^2 = 25 \] \[ DC = \sqrt{25} = 5 \]
Thus, the length of \( DC \) is 5.
Therefore, the correct answer is: 5.
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