1. **State the problem:** We are given a triangle with area $x^2$ square centimeters, base length $2x + 22.5$ cm, and height $x - 5$ cm. We need to find the value of $x$. 2. **Formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Set up the equation using given values:** $$x^2 = \frac{1}{2} \times (2x + 22.5) \times (x - 5)$$ 4. **Multiply both sides by 2 to eliminate the fraction:** $$2x^2 = (2x + 22.5)(x - 5)$$ 5. **Expand the right side:** $$2x^2 = 2x \times x - 2x \times 5 + 22.5 \times x - 22.5 \times 5$$ $$2x^2 = 2x^2 - 10x + 22.5x - 112.5$$ 6. **Simplify the right side:** $$2x^2 = 2x^2 + 12.5x - 112.5$$ 7. **Subtract $2x^2$ from both sides:** $$2x^2 - \cancel{2x^2} = \cancel{2x^2} + 12.5x - 112.5 - 2x^2$$ $$0 = 12.5x - 112.5$$ 8. **Solve for $x$:** $$12.5x = 112.5$$ $$x = \frac{112.5}{12.5}$$ $$x = 9$$ 9. **Check the height is positive:** $$x - 5 = 9 - 5 = 4 > 0$$ so the solution is valid. **Final answer:** $$\boxed{9}$$