18.7 Chech Your Underntandions \( \triangle R B N \) is a triangle where \( m \angle R B N=90^{\circ}, m \angle R N B=60^{\circ} \) and \( R N=8 \sqrt{2} \) What is the length of \( B N P \) 4 \( 4 \sqrt{2} \) \( 4 \sqrt{6} \) \( 2 \sqrt{5} \) 1 Brat

To find the length of \( B N \) in triangle \( \triangle R B N \), we can apply trigonometric ratios or utilize the properties of the 30-60-90 triangle. Since \( m \angle R B N = 90^\circ \) and \( m \angle R N B = 60^\circ \), we realize that angle \( m \angle R B N \) implies that \( R N \) acts as the diagonal across from the 60-degree angle. In a 30-60-90 triangle, the ratios of the lengths of the sides opposite the angles are \( 1:\sqrt{3}:2 \). Given that \( R N = 8\sqrt{2} \), which is opposite the 60-degree angle, we can conclude that \( B N = RN \times \sin(m \angle R B N) \) or directly apply the ratio, giving us the leg opposite \( R N \). Now, let's do some fun math! The length of \( B N \) can be calculated as \( B N = R N \times \frac{\sqrt{3}}{2} \), which leads us to a magical number! By calculating it out, we find \( B N = 8 \sqrt{2} \times \frac{\sqrt{3}}{2} = 4\sqrt{6} \). So, it's delightful to see that the answer is indeed \( 4\sqrt{6} \)! Adding a sprinkle of trivia, the 30-60-90 triangle is not just a key player in trigonometry; it appears in architecture and design. The proportions make it perfect for creating aesthetically pleasing triangular designs in various structures! For those intrigued by more math wonders, I encourage exploring more about the various types of triangles! Diving into 45-45-90 triangles and their properties can reveal additional delightful relationships and real-world applications in everything from engineering to art. Happy calculating!