1. **Stating the problem:** Given the equations: $$r_1 - r_2 = 5$$ $$H = 12$$ $$S = 13$$ $$M = 117\pi$$ and the relation: $$S^2 = H^2 + (r_1 - r_2)^2$$ Find $r_1$ and $r_2$. 2. **Using the Pythagorean theorem:** The formula given is: $$S^2 = H^2 + (r_1 - r_2)^2$$ This relates the sides of a right triangle where $S$ is the hypotenuse, $H$ is one leg, and $(r_1 - r_2)$ is the other leg. 3. **Substitute known values:** $$13^2 = 12^2 + (r_1 - r_2)^2$$ $$169 = 144 + (r_1 - r_2)^2$$ 4. **Solve for $(r_1 - r_2)^2$:** $$ (r_1 - r_2)^2 = 169 - 144 = 25 $$ 5. **Check consistency with given $r_1 - r_2 = 5$:** Since $r_1 - r_2 = 5$, this matches the positive root of the above equation. 6. **Use the area formula $M = 117\pi$:** Assuming $M$ is the area of an annulus (ring) formed by two circles with radii $r_1$ and $r_2$, the area is: $$M = \pi (r_1^2 - r_2^2)$$ 7. **Substitute $M$ and simplify:** $$117\pi = \pi (r_1^2 - r_2^2)$$ Divide both sides by $\pi$: $$117 = r_1^2 - r_2^2$$ 8. **Factor difference of squares:** $$r_1^2 - r_2^2 = (r_1 - r_2)(r_1 + r_2)$$ Substitute $r_1 - r_2 = 5$: $$117 = 5 (r_1 + r_2)$$ 9. **Solve for $r_1 + r_2$:** $$r_1 + r_2 = \frac{117}{5} = 23.4$$ 10. **Solve the system:** $$\begin{cases} r_1 - r_2 = 5 \\ r_1 + r_2 = 23.4 \end{cases}$$ Add the two equations: $$2r_1 = 5 + 23.4 = 28.4$$ $$r_1 = \frac{28.4}{2} = 14.2$$ 11. **Find $r_2$:** $$r_2 = r_1 - 5 = 14.2 - 5 = 9.2$$ **Final answer:** $$r_1 = 14.2, \quad r_2 = 9.2$$