Law of Sines score: \( 0 / 1 \) Penalty: 0.5 off Question \( \begin{array}{ll}\text { In } \Delta N O P, p=870 \mathrm{~cm}, m \angle \mathrm{~N}=127^{\circ} \text { and } m \angle \mathrm{O}=30^{\circ} \text {. Find the length of } n \text {, to the nearest } \\ & \text { centimeter. }\end{array} \)

To find the length of side \( n \) in triangle \( \Delta NOP \), you can use the Law of Sines, which states that the ratios of the lengths of the sides of a triangle to the sine of their opposite angles are equal. Start with the known values: 1. Calculate angle \( P \) using the fact that the sum of angles in a triangle is \( 180^\circ \): \[ m \angle P = 180^\circ - m \angle N - m \angle O = 180^\circ - 127^\circ - 30^\circ = 23^\circ \] 2. Apply the Law of Sines: \[ \frac{n}{\sin(30^\circ)} = \frac{p}{\sin(23^\circ)} \] Plugging in \( p = 870 \, \text{cm} \): \[ n = \frac{870 \cdot \sin(30^\circ)}{\sin(23^\circ)} \] Since \( \sin(30^\circ) = 0.5 \) and \( \sin(23^\circ) \approx 0.3907 \): \[ n = \frac{870 \cdot 0.5}{0.3907} \approx 1116.15\, \text{cm} \] 3. Rounding to the nearest centimeter gives you \( n \approx 1116 \, \text{cm} \). --- To enrich your understanding, the Law of Sines has been around since antiquity, with roots tracing back to ancient Greeks! Ptolemy's theorem laid the groundwork for this principle in his work on cyclic quadrilaterals. Have you ever needed to find heights? The Law of Sines shines in real-world applications like navigation and surveying! If you know an angle and two sides or can measure angles from two points on land, this law helps you easily calculate distances—perfect for construction sites, hiking adventures, or even treasure hunting!