1. **State the problem:** We are given the cosine of angle A in a triangle with vertices A, B, and C, and the formula for cosine of angle A is given by the law of cosines: $$\cos A = \frac{AB^2 + AC^2 - BC^2}{2 \cdot AB \cdot AC}$$ We know that $\cos A = -\frac{1}{5}$. 2. **Explain the formula:** The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful for finding an angle when the lengths of all three sides are known, or vice versa. 3. **Interpret the given value:** Since $\cos A = -\frac{1}{5}$, angle A is obtuse because cosine is negative for angles between 90° and 180°. 4. **Use the formula to find relationships:** $$-\frac{1}{5} = \frac{AB^2 + AC^2 - BC^2}{2 \cdot AB \cdot AC}$$ Multiply both sides by $2 \cdot AB \cdot AC$: $$-\frac{1}{5} \times 2 \cdot AB \cdot AC = AB^2 + AC^2 - BC^2$$ $$\Rightarrow -\frac{2}{5} AB AC = AB^2 + AC^2 - BC^2$$ 5. **Rearrange to express $BC^2$:** $$BC^2 = AB^2 + AC^2 + \frac{2}{5} AB AC$$ This equation relates the side lengths of the triangle given the cosine of angle A. 6. **Summary:** Given $\cos A = -\frac{1}{5}$, the side lengths satisfy: $$BC^2 = AB^2 + AC^2 + \frac{2}{5} AB AC$$ This can be used to find one side if the other two are known, or to check consistency of side lengths in the triangle.