To solve the problem (assuming the common setup of two circles with radii 5 and 13, distance between centers 12), follow these steps:

Key Observations

Let the circles have centers (A) (radius 5) and (B) (radius 13), with (AB = 12). Let the common chord be (CD), and (O) be the midpoint of (CD) (so (AB \perp CD) and (CO = OD)).

Equations for Right Triangles

For (\triangle AOC):
(AO^2 + CO^2 = 5^2) ...(1)

For (\triangle BOC):
((12 - AO)^2 + CO^2 = 13^2) ...(2)

Subtract (1) from (2)

((12 - AO)^2 - AO^2 = 13^2 - 5^2)
(144 - 24AO + AO^2 - AO^2 = 169 - 25)
(144 - 24AO = 144)
(AO = 0)

Calculate Chord Length

Since (AO = 0), (O = A). Then from (1):
(0 + CO^2 = 25) → (CO = 5)

Thus, (CD = 2 \times CO = 10).

Answer: (\boxed{10})

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