In a rhombus, the diagonals bisect each other at right angles and also bisect the opposite angles. Given the segments \(AD\) and \(DB\), we can find the length of \(DC\) using the information provided.
Since \(AD = 13\) and \(DB = 24\), point \(D\) is the intersection of the diagonals, creating two triangles \(ABD\) and \(BDC\). Because the diagonals bisect each other, \(DC\) is equal to \(DB\), and \(AC\) is equal to \(AD\).
In a rhombus:
- \(AC\) and \(BD\) are the diagonals.
- Since \(D\) is the midpoint of both diagonals, we have:
\[ AD + DC = AC \quad \text{and} \quad DB + BD = AB \]
We know:
- \(AD = 13\)
- \(DB = 24\)
To find \(DC\), we note that in triangle \(ABD\), by using the Pythagorean Theorem:
\[ AB^2 = AD^2 + DB^2 \]
However, we don't need to find \(AB\) yet; we can use the properties of the rhombus diagonals directly.
Since \(DB\) is divided into two equal parts by the diagonal \(AC\), and since \(DB\) measures \(24\), we can say:
\[ DC = DB \]
However, we can also calculate it by using properties of triangles formed by the diagonals. Let's define:
- Let's call the intersection point of the diagonals \(O\).
Then:
- \(AO = \frac{AC}{2}\)
- \(DO = \frac{DB}{2}\)
With: \[ DO = \frac{24}{2} = 12 \]
Therefore:
- Since \(DC\) is opposite to \(DB\) in triangle \(BDC\), and \(DO\) is perpendicular:
By the argument of CPCTC: \[ DC = DB = 12 \]
After verifying no triangle properties needed adjustment, we find:
The length of \(DC\) is 12.
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