In a rhombus, all sides are equal in length. Since we know that \( AD = 13 \) and \( DB = 24 \), and given that \( D \) is a vertex of the rhombus, we can use the lengths of \( AD \) and \( DB \) to find the length of the diagonal \( DC \).
First, observe that in a rhombus, the diagonals bisect each other at right angles. In a rhombus, let’s denote the vertices as \( A, B, C, \) and \( D \) such that:
- \( A \) and \( C \) are one pair of opposite vertices,
- \( B \) and \( D \) are the other pair.
Since \( AD \) and \( DB \) are parts of the diagonals:
- Let \( O \) be the point where the two diagonals intersect, which means \( O \) will be the midpoint of both diagonals.
From the properties of a rhombus, we can set:
- \( AO = OC \)
- \( BO = OD \)
Thus, since \( A \) and \( D \) bisect each other, we can set:
\( AD = AO + OD \)
Given that:
- \( AO = OD = \frac{AD}{2} = \frac{13}{2} = 6.5 \)
- Since \( BO = OC = \frac{DB}{2} = \frac{24}{2} = 12 \)
Now, to find the length of \( DC \):
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Calculating the full lengths based on the bisected parts:
- From the midpoint \( O \), since \( AO + DC = AC \),
- We have already found \( AO = 6.5 \) (or \( 13/2 \)) and \( OC = 12 \).
Thus, we can find \( AC \) by calculating:
\[ AC = AO + OC = 6.5 + 12 = 18.5 \]
Now, calculate the lengths of both diagonals:
Based on the diagonals:
- The length of \( DC = OC = 12 \), because \( AC \) is comprised of the lengths \( AO \) and \( OC \):
Thus the final length of \( DC \) is \( 12 \).
So, the length of \( DC \) is 12.
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