To find the length of \( DC \) in a rhombus \( ABCD \) with diagonals \( AC \) and \( BD \) intersecting at point \( O \), we need to use some properties of rhombuses and symmetry.
Given:
- \( AD = 13 \)
- \( DB = 24 \)
Since \( ABCD \) is a rhombus, all its sides are equal, which means \( AB = BC = CD = DA \).
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Finding the Length of \( AB \): \[ AB = AD = 13 \]
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Using the Properties of the Diagonals: The diagonals of a rhombus bisect each other at right angles. Therefore, we have: \[ AO = OC \quad \text{and} \quad BO = OD \]
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Given Information: Since \( DB = 24 \), and \( O \) is the midpoint of \( DB \): \[ BO = \frac{DB}{2} = \frac{24}{2} = 12 \] Thus, \( OD = 12 \) as well since \( O \) bisects \( DB \).
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Triangle \( AOD \): Now, we can apply the Pythagorean theorem in triangle \( AOD \): \[ AD^2 = AO^2 + OD^2 \] Plugging in the values: \[ 13^2 = AO^2 + 12^2 \] Simplifying: \[ 169 = AO^2 + 144 \] Subtracting 144 from both sides: \[ AO^2 = 25 \] Therefore: \[ AO = 5 \] Since \( AC \) is bisected by \( O \): \[ OC = AO = 5 \]
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Finding Length of \( AC \): The entire diagonal \( AC \) will be: \[ AC = AO + OC = 5 + 5 = 10 \]
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Finding Length of \( DC \): In a rhombus, the diagonals bisect the angles. We also know that \( AC \) and \( BD \) are perpendicular bisectors. Using the congruence of triangles \( AOB \) and \( COD \) yields: \[ AD = DC \]
Since \( AC = 10 \) is composed of components \( AO \) and \( OC \), we do not need to calculate \( DC \) as it matches the sides of the rhombus.
Thus, \( DC = AD = 13 \).
In conclusion, the length of \( DC \) in the rhombus is \( \boxed{13} \).
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