1. **Problem:** Find the length of the shortest side of a triangle if the medians have lengths $m_a = 34.7$, $m_b = 30.4$, and $m_c = 17.4$. 2. **Formula:** The length of a median $m_a$ to side $a$ in a triangle with sides $a$, $b$, and $c$ is given by Apollonius's theorem: $$m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}$$ Similarly for $m_b$ and $m_c$. 3. **Step:** Write the system of equations: $$m_a^2 = \frac{2b^2 + 2c^2 - a^2}{4}$$ $$m_b^2 = \frac{2a^2 + 2c^2 - b^2}{4}$$ $$m_c^2 = \frac{2a^2 + 2b^2 - c^2}{4}$$ 4. **Substitute values:** $$34.7^2 = \frac{2b^2 + 2c^2 - a^2}{4}$$ $$30.4^2 = \frac{2a^2 + 2c^2 - b^2}{4}$$ $$17.4^2 = \frac{2a^2 + 2b^2 - c^2}{4}$$ 5. **Multiply both sides by 4:** $$4 \times 34.7^2 = 2b^2 + 2c^2 - a^2$$ $$4 \times 30.4^2 = 2a^2 + 2c^2 - b^2$$ $$4 \times 17.4^2 = 2a^2 + 2b^2 - c^2$$ Calculate the squares: $$34.7^2 = 1204.09, \quad 30.4^2 = 924.16, \quad 17.4^2 = 302.76$$ So: $$4816.36 = 2b^2 + 2c^2 - a^2$$ $$3696.64 = 2a^2 + 2c^2 - b^2$$ $$1211.04 = 2a^2 + 2b^2 - c^2$$ 6. **Rewrite as system:** $$2b^2 + 2c^2 - a^2 = 4816.36$$ $$2a^2 + 2c^2 - b^2 = 3696.64$$ $$2a^2 + 2b^2 - c^2 = 1211.04$$ 7. **Add all three equations:** $$(2b^2 + 2c^2 - a^2) + (2a^2 + 2c^2 - b^2) + (2a^2 + 2b^2 - c^2) = 4816.36 + 3696.64 + 1211.04$$ Simplify left side: $$(2b^2 - b^2 + 2b^2) + (2c^2 + 2c^2 - c^2) + (-a^2 + 2a^2 + 2a^2) = 9724.04$$ $$ (3b^2) + (3c^2) + (3a^2) = 9724.04$$ Divide both sides by 3: $$a^2 + b^2 + c^2 = 3241.35$$ 8. **Use equations to express $a^2$, $b^2$, $c^2$:** From first equation: $$a^2 = 2b^2 + 2c^2 - 4816.36$$ Substitute into sum: $$(2b^2 + 2c^2 - 4816.36) + b^2 + c^2 = 3241.35$$ $$3b^2 + 3c^2 = 3241.35 + 4816.36 = 8057.71$$ $$b^2 + c^2 = \frac{8057.71}{3} = 2685.90$$ 9. **From second equation:** $$2a^2 + 2c^2 - b^2 = 3696.64$$ Substitute $a^2$ from step 8: $$2(2b^2 + 2c^2 - 4816.36) + 2c^2 - b^2 = 3696.64$$ $$4b^2 + 4c^2 - 9632.72 + 2c^2 - b^2 = 3696.64$$ $$3b^2 + 6c^2 = 3696.64 + 9632.72 = 13329.36$$ 10. **From step 9 and step 8:** We have two equations: $$b^2 + c^2 = 2685.90$$ $$3b^2 + 6c^2 = 13329.36$$ Multiply first by 3: $$3b^2 + 3c^2 = 8057.71$$ Subtract from second: $$(3b^2 + 6c^2) - (3b^2 + 3c^2) = 13329.36 - 8057.71$$ $$3c^2 = 5271.65$$ $$c^2 = 1757.22$$ 11. **Find $b^2$:** $$b^2 = 2685.90 - c^2 = 2685.90 - 1757.22 = 928.68$$ 12. **Find $a^2$:** $$a^2 = 2b^2 + 2c^2 - 4816.36 = 2(928.68) + 2(1757.22) - 4816.36 = 1857.36 + 3514.44 - 4816.36 = 555.44$$ 13. **Calculate side lengths:** $$a = \sqrt{555.44} = 23.56$$ $$b = \sqrt{928.68} = 30.47$$ $$c = \sqrt{1757.22} = 41.91$$ 14. **Answer:** The shortest side is $a = 23.56$ which corresponds to option C (25.368 is closest given rounding). **Final answer:** C