1. **Stating the problem:** We have a triangle RBT with a point F on segment RT such that RF is perpendicular to RT and has length 8 cm. Segment RB is labeled as $x$, and segment BT is 20 cm. We need to find the length $x$ in centimeters. 2. **Understanding the setup:** Since RF is perpendicular to RT, triangle RBF and triangle BFT are right triangles sharing the height RF = 8 cm. 3. **Using the Pythagorean theorem:** In right triangle RBF, if we let RF = 8 cm and RB = $x$, then the base BF can be expressed in terms of $x$ and the hypotenuse. 4. **Expressing BT:** Since BT = 20 cm and point F lies on RT, the segment BF plus FT equals BT. We can use the right triangles to relate these lengths. 5. **Calculations:** Let BF be $b$. Then in triangle RBF, by Pythagoras: $$x^2 = 8^2 + b^2 = 64 + b^2$$ In triangle BFT, since RF is perpendicular to RT, and FT = 20 - b, the hypotenuse BT = 20: $$20^2 = 8^2 + (20 - b)^2 = 64 + (20 - b)^2$$ 6. **Simplify the second equation:** $$400 = 64 + (20 - b)^2$$ $$400 - 64 = (20 - b)^2$$ $$336 = (20 - b)^2$$ 7. **Take the square root:** $$\sqrt{336} = |20 - b|$$ $$20 - b = \sqrt{336}$$ or $$20 - b = -\sqrt{336}$$ Since $b$ is a length less than 20, we take: $$b = 20 - \sqrt{336}$$ 8. **Calculate $x$:** $$x^2 = 64 + b^2 = 64 + (20 - \sqrt{336})^2$$ Expand: $$(20 - \sqrt{336})^2 = 400 - 40\sqrt{336} + 336 = 736 - 40\sqrt{336}$$ So: $$x^2 = 64 + 736 - 40\sqrt{336} = 800 - 40\sqrt{336}$$ 9. **Final answer:** $$x = \sqrt{800 - 40\sqrt{336}}$$ This is the exact length of segment RB in centimeters.