We are given a triangle $ABC$ with an altitude $BD$. We know that $|AB|=13$, $|BD|=12$, and $|DC|=4$. We need to calculate the lengths of $|AC|$ and $|BC|$.

1. Problem Description

We are given a triangle $ABC$ with an altitude $BD$. We know that $|AB|=13$, $|BD|=12$, and $|DC|=4$. We need to calculate the lengths of $|AC|$ and $|BC|$.

2. Solution Steps

First, we consider the right triangle $ABD$. By the Pythagorean theorem, we have

$|AB|^2 = |AD|^2 + |BD|^2$

$13^2 = |AD|^2 + 12^2$

$169 = |AD|^2 + 144$

$|AD|^2 = 169 - 144 = 25$

$|AD| = \sqrt{25} = 5$.

Now we can find $|AC|$ by adding $|AD|$ and $|DC|$.

$|AC| = |AD| + |DC| = 5 + 4 = 9$.

Next, we consider the right triangle $BCD$. By the Pythagorean theorem, we have

$|BC|^2 = |BD|^2 + |DC|^2$

$|BC|^2 = 12^2 + 4^2$

$|BC|^2 = 144 + 16 = 160$

$|BC| = \sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}$.

3. Final Answer

$|AC| = 9$

$|BC| = 4\sqrt{10}$