1. **State the problem:** We have two triangles, P and Q, where Q is a scaled copy of P. The side lengths of triangle P are 9, 6, and 12. The corresponding side lengths of triangle Q are 12, $x$, and 16. We need to find the value of $x$. 2. **Understand the concept:** Since triangle Q is a scaled copy of triangle P, their corresponding sides are proportional. This means the ratios of corresponding sides are equal. 3. **Set up the proportion:** The side of length 9 in P corresponds to 12 in Q, the side of length 6 in P corresponds to $x$ in Q, and the side of length 12 in P corresponds to 16 in Q. 4. **Write the ratios:** $$\frac{9}{12} = \frac{6}{x} = \frac{12}{16}$$ 5. **Simplify the known ratios:** $$\frac{9}{12} = \frac{3}{4}$$ $$\frac{12}{16} = \frac{3}{4}$$ Both simplify to $\frac{3}{4}$, confirming the scale factor is $\frac{3}{4}$. 6. **Use the proportion to find $x$:** $$\frac{6}{x} = \frac{3}{4}$$ 7. **Solve for $x$:** Multiply both sides by $x$: $$6 = \frac{3}{4} x$$ Multiply both sides by 4 to clear the denominator: $$4 \times 6 = 4 \times \frac{3}{4} x$$ $$24 = 3x$$ Divide both sides by 3: $$\frac{24}{\cancel{3}} = \frac{3x}{\cancel{3}}$$ $$8 = x$$ 8. **Final answer:** $$x = 8$$ Therefore, the value of $x$ is 8.