1. **State the problem:** We are given a triangle ABC with lengths AB = $3x - 4$, AC = $2x + 12$, and BC = $7x - 2$. It is given that the ratio $AB : AC = 1 : 2$. We need to show that $AC : BC = 2 : 3$. 2. **Express the given ratio mathematically:** Given $AB : AC = 1 : 2$ implies $$\frac{AB}{AC} = \frac{1}{2}$$ Substitute the lengths: $$\frac{3x - 4}{2x + 12} = \frac{1}{2}$$ 3. **Solve for $x$:** Cross multiply: $$2(3x - 4) = 1(2x + 12)$$ $$6x - 8 = 2x + 12$$ Subtract $2x$ from both sides: $$6x - 2x - 8 = 12$$ $$4x - 8 = 12$$ Add 8 to both sides: $$4x = 20$$ Divide both sides by 4: $$x = 5$$ 4. **Find the actual lengths of AC and BC:** $$AC = 2x + 12 = 2(5) + 12 = 10 + 12 = 22$$ $$BC = 7x - 2 = 7(5) - 2 = 35 - 2 = 33$$ 5. **Find the ratio $AC : BC$ and simplify:** $$AC : BC = 22 : 33$$ Divide both terms by 11: $$2 : 3$$ 6. **Conclusion:** We have shown that $AC : BC = 2 : 3$, as required.