1. **State the problem:** We are given a triangle with sides $x$, $x+2$, and $x+4$ and a perimeter of 24 cm. We need to find the value of $x$ and determine the type of triangle. 2. **Form the equation for the perimeter:** The perimeter $P$ is the sum of all sides: $$x + (x+2) + (x+4) = 24$$ 3. **Simplify the equation:** $$3x + 6 = 24$$ 4. **Isolate $x$:** $$3x = 24 - 6$$ $$3x = 18$$ 5. **Divide both sides by 3:** $$\cancel{3}x = \frac{18}{\cancel{3}}$$ $$x = 6$$ 6. **Find the side lengths:** $$x = 6, \quad x+2 = 8, \quad x+4 = 10$$ 7. **Determine the type of triangle:** Check if it is a right triangle using the Pythagorean theorem: $$6^2 + 8^2 = 36 + 64 = 100$$ $$10^2 = 100$$ Since $$6^2 + 8^2 = 10^2$$, the triangle is a right triangle (Pythagorean triple). **Final answer:** $$x = 6$$ The triangle with sides 6, 8, and 10 is a right triangle.