Solution: The angles are $30^\circ$, $60^\circ$, and $90^\circ$. Using the Law of Sines: $\frac{5}{\sin 30^\circ} = \frac{c}{\sin 90^\circ}$. Since $\sin 30^\circ = 0.5$ and $\sin 90^\circ = 1$, $c = \frac{5}{0.5} = 10$ km.
["Understanding Right Triangle Geometry: Solving for Side $c$ Using the Law of Sines", "In triangle geometry, certain angle measures reveal powerful relationships that help solve for unknown side lengths. One of the most foundational tools in this context is the Law of Sines, a formula that relates the lengths of a triangle’s sides to the sines of its opposite angles:", "[\n\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\n]", "This article explores a classic right triangle problem involving angles of $30^\circ$, $60^\circ$, and $90^\circ$, demonstrating how to apply the Law of Sines to determine the length of the hypotenuse.", "---", "### The Special Right Triangle: $30^\circ$-$60^\circ$-$90^\circ$", "A $30^\circ$-$60^\circ$-$90^\circ$ triangle is a special right triangle with well-known side ratios. In this triangle:", "- The angles are $30^\circ$, $60^\circ$, and $90^\circ$.\n- The sides opposite these angles have consistent length ratios:\nOpposite $30^\circ$ → shortest side (ratio 1),\nOpposite $60^\circ$ → middle side (ratio $\sqrt{3}$),\nOpposite $90^\circ$ (the hypotenuse) → longest side (ratio 2).", "While trigonometric ratios like sine and cosine simplify calculations, the Law of Sines provides an alternative and elegant verification method, especially when dealing with non-right triangles or reinforcing triangle properties.", "---", "### Applying the Law of Sines", "Given:", "- $\angle A = 30^\circ$, side $a = 5$ km\n- $\angle B = 60^\circ$\n- $\angle C = 90^\circ$, so $\sin 90^\circ = 1$", "Using the Law of Sines:", "[\n\frac{a}{\sin A} = \frac{c}{\sin C}\n]", "Substitute known values:", "[\n\frac{5}{\sin 30^\circ} = \frac{c}{\sin 90^\circ}\n]", "We know:", "- $\sin 30^\circ = 0.5$\n- $\sin 90^\circ = 1$", "Thus:", "[\n\frac{5}{0.5} = \frac{c}{1}\n]", "Simplify:", "[\n10 = c\n]", "Therefore, the length of side $c$ — the hypotenuse — is 10 km.", "---", "### Why This Matters: Precision and Verification", "Using the Law of Sines allows you to independently confirm triangle measurements and side lengths without relying solely on trigonometric ratios. This method strengthens your geometric reasoning and ensures accuracy, particularly in complex problems involving multiple triangles or non-right-angled configurations.", "---", "### Summary", "- The $30^\circ$-$60^\circ$-$90^\circ$ triangle follows a precise side ratio linked to its angles.\n- The Law of Sines offers a direct way to compute unknown sides using sine values of opposite angles.\n- Applying $\frac{5}{\sin 30^\circ} = \frac{c}{\sin 90^\circ}$ yields $c = 10$ km, confirming the hypotenuse in a right triangle.\n- Mastering these techniques builds a solid foundation for advanced geometry and trigonometry applications.", "Whether you’re solving textbook problems or tackling real-world spatial challenges, understanding how angles and side lengths interrelate using the Law of Sines is an essential skill in mathematics and engineering fields.", "---", "Key takeaway: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle with a side opposite $30^\circ$ measuring 5 km, the hypotenuse is always twice that length—easily found via $\frac{5}{\sin 30^\circ} = \frac{c}{\sin 90^\circ}$—confirming $c = 10$ km."]
