1. **State the problem:** We have a right triangle EFG with a right angle at F. A perpendicular segment FH is dropped from F to the base EG. Given EH = 14 and GH = 6, we need to find FH. 2. **Understand the setup:** Since FH is perpendicular to EG, triangle EFH and triangle GFH are right triangles sharing the height FH. 3. **Use the geometric mean theorem (right triangle altitude theorem):** The altitude to the hypotenuse in a right triangle satisfies: $$FH^2 = EH \times GH$$ 4. **Substitute the given values:** $$FH^2 = 14 \times 6 = 84$$ 5. **Solve for FH:** $$FH = \sqrt{84}$$ 6. **Simplify the square root:** $$\sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}$$ 7. **Calculate the decimal value:** $$FH \approx 2 \times 4.5826 = 9.165\approx 9.17$$ **Final answer:** $$FH \approx 9.17$$