1. **State the problem:** We need to find the sine of angle $F$ in the first right triangle $FHG$. 2. **Analyze the given information:** The second triangle $JKI$ is a right triangle with sides $KJ=13$, $KI=5$, and $IJ=12$. Since $JKI$ is a right triangle, these side lengths satisfy the Pythagorean theorem: $$5^2 + 12^2 = 25 + 144 = 169 = 13^2.$$ This confirms the triangle is right-angled at $I$. 3. **Relate the two triangles:** The problem implies the first triangle $FHG$ is similar to the second triangle $JKI$ because both are right triangles and the labeling suggests corresponding angles. 4. **Identify sides relative to angle $F$ in triangle $FHG$:** In triangle $FHG$, angle $F$ is at vertex $F$. The side opposite angle $F$ is $HG$, the side adjacent to angle $F$ is $FG$, and the hypotenuse is $FH$. 5. **Use similarity to find side ratios:** Since $JKI$ has sides $5$ (adjacent to $K$), $12$ (opposite to $K$), and $13$ (hypotenuse), and angle $F$ corresponds to angle $K$, the ratio of opposite to hypotenuse for angle $F$ is the same as for angle $K$: $$\sin(F) = \sin(K) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}.$$ 6. **Final answer:** $$\sin(F) = \frac{12}{13}.$$