Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

Key Concepts

Bearings

Bearings are a way of describing direction using angles measured clockwise from the north. In this problem, the bearings of the tree from both Mark and Lisa are given, which helps in determining the relative positions of the points involved. Understanding how to interpret and use bearings is crucial for solving problems involving angles and distances in navigation and surveying.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this scenario, the angles formed by the bearings can be used to set up right triangles, allowing the application of these ratios to find unknown distances. Mastery of these ratios is essential for solving problems involving triangles in trigonometry.

Parallel Lines and Transversals

When two lines are parallel, any line that crosses them creates corresponding angles that are equal. In this case, the banks of the river are parallel, and the bearings create angles that can be analyzed using properties of parallel lines. Recognizing these relationships is important for establishing the geometric framework needed to calculate distances across the river.