1. **State the problem:** We have a right triangle with sides labeled as $x + 2$ cm, $\frac{x}{2} + 4$ cm, and $x + 5$ cm. The perimeter is given as 36 cm. We need to find the value of $x$. 2. **Write the perimeter formula:** The perimeter $P$ of a triangle is the sum of the lengths of its sides: $$P = \text{side}_1 + \text{side}_2 + \text{side}_3$$ 3. **Set up the equation:** Substitute the given side lengths and perimeter: $$x + 2 + \left(\frac{x}{2} + 4\right) + (x + 5) = 36$$ 4. **Simplify the equation:** Combine like terms: $$x + 2 + \frac{x}{2} + 4 + x + 5 = 36$$ $$\left(x + \frac{x}{2} + x\right) + (2 + 4 + 5) = 36$$ $$\left(2x + \frac{x}{2}\right) + 11 = 36$$ 5. **Combine the $x$ terms:** $$2x + \frac{x}{2} = \frac{4x}{2} + \frac{x}{2} = \frac{5x}{2}$$ 6. **Rewrite the equation:** $$\frac{5x}{2} + 11 = 36$$ 7. **Isolate $x$:** Subtract 11 from both sides: $$\frac{5x}{2} = 36 - 11$$ $$\frac{5x}{2} = 25$$ 8. **Solve for $x$:** Multiply both sides by 2: $$5x = 25 \times 2$$ $$5x = 50$$ 9. **Divide both sides by 5:** $$x = \frac{50}{5}$$ $$x = 10$$ **Final answer:** $x = 10$ cm.