1. **State the problem:** We have two forces acting at right angles (90°) with a resultant of 10 kN, and the same two forces acting at an angle of 60° with a resultant of 12 kN. We need to find the magnitudes of the two forces, say $F_1$ and $F_2$. 2. **Formula for resultant of two forces:** When two forces $F_1$ and $F_2$ act at an angle $\theta$, the magnitude of the resultant $R$ is given by: $$ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} $$ 3. **Apply the formula for right angle (90°):** Since $\cos 90^\circ = 0$, the resultant is: $$ 10 = \sqrt{F_1^2 + F_2^2} $$ Square both sides: $$ 10^2 = F_1^2 + F_2^2 $$ $$ 100 = F_1^2 + F_2^2 \quad \quad (1) $$ 4. **Apply the formula for 60° angle:** $$ 12 = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos 60^\circ} $$ Since $\cos 60^\circ = 0.5$: $$ 12 = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \times 0.5} $$ Simplify: $$ 12 = \sqrt{F_1^2 + F_2^2 + F_1 F_2} $$ Square both sides: $$ 12^2 = F_1^2 + F_2^2 + F_1 F_2 $$ $$ 144 = F_1^2 + F_2^2 + F_1 F_2 \quad \quad (2) $$ 5. **Subtract equation (1) from (2):** $$ 144 - 100 = (F_1^2 + F_2^2 + F_1 F_2) - (F_1^2 + F_2^2) $$ $$ 44 = F_1 F_2 $$ 6. **We now have two equations:** $$ F_1^2 + F_2^2 = 100 $$ $$ F_1 F_2 = 44 $$ 7. **Solve for $F_1$ and $F_2$:** Let $x = F_1$ and $y = F_2$. From $x y = 44$, express $y = \frac{44}{x}$. Substitute into $x^2 + y^2 = 100$: $$ x^2 + \left(\frac{44}{x}\right)^2 = 100 $$ $$ x^2 + \frac{1936}{x^2} = 100 $$ Multiply both sides by $x^2$: $$ x^4 + 1936 = 100 x^2 $$ Rewrite: $$ x^4 - 100 x^2 + 1936 = 0 $$ 8. **Let $z = x^2$, then:** $$ z^2 - 100 z + 1936 = 0 $$ 9. **Solve quadratic for $z$:** $$ z = \frac{100 \pm \sqrt{100^2 - 4 \times 1936}}{2} $$ Calculate discriminant: $$ 10000 - 7744 = 2256 $$ $$ \sqrt{2256} \approx 47.5 $$ So: $$ z = \frac{100 \pm 47.5}{2} $$ Two solutions: $$ z_1 = \frac{100 + 47.5}{2} = 73.75 $$ $$ z_2 = \frac{100 - 47.5}{2} = 26.25 $$ 10. **Find $x$ values:** $$ x = \sqrt{z} $$ $$ x_1 = \sqrt{73.75} \approx 8.59 $$ $$ x_2 = \sqrt{26.25} \approx 5.12 $$ 11. **Find corresponding $y$ values:** $$ y_1 = \frac{44}{8.59} \approx 5.12 $$ $$ y_2 = \frac{44}{5.12} \approx 8.59 $$ 12. **Final answer:** The magnitudes of the two forces are approximately $8.59$ kN and $5.12$ kN.