Annie is in a kayak in the middle of a channel between Orcas...

Annie is in a kayak in the middle of a channel between Orcas Island and Blakely Island, two of the San Juan Islands in Washington State. She knows from readings she has taken in the past that if she considers the town of Deer Harbor to be at the origin, then the summit of Constitution Peak is at $(7.9,4.0)$, while the summit of Blakely Peak is at $(9.3,-3.4)$. All distances are given in miles. She pulls out her compass and finds that the summit of Constitution Peak is 15 degrees east of north from her, while the summit of Blakely Peak is at 100 degrees from north. How far is Annie from Deer Harbor?

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with common variables. In many geometric problems, including those involving bearings and coordinates, forming a system of equations allows one to simultaneously consider multiple directional constraints. Solving the system provides the coordinates of an unknown point that satisfies all the conditions, such as the intersection of lines defined by different bearing directions.

Distance Formula

The distance formula is a method for calculating the straight-line distance between two points in a coordinate plane. Based on the Pythagorean theorem, it computes the distance by taking the square root of the sum of the squares of the differences in corresponding coordinates. This formula is fundamental when determining how far an object is from a reference point after its coordinates have been identified.

Line Equations and Slopes

This concept deals with representing directional paths as linear equations in the coordinate plane. Given an angle of direction, one can compute the slope of the corresponding line using the tangent function. These linear equations are essential for describing the paths from an unknown point towards known landmarks, thus enabling the determination of intersection points that reveal the unknown location.

Trigonometric Relationships

Trigonometric relationships connect angles with ratios of sides in right triangles. In problems involving bearings and distances, sine, cosine, and tangent functions are used to convert between angular directions and linear components. This conversion allows the formulation of equations that describe the directional paths or positions of points relative to a coordinate system.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. By assigning coordinates to points in the plane, perhaps with a fixed origin, one can precisely describe the location of objects and use algebraic methods to solve geometric problems, such as determining distances and intersections between points and lines.

Compass Bearings and Angle Conversions

This concept involves understanding how directions given as compass bearings (for example, degrees east of north or measured from north) relate to angles in a standard coordinate system. It requires converting these bearings into angles that can be used in trigonometric functions, typically by relating them to the positive x?axis or y?axis, which is essential for mapping the directions of objects or points in a plane.

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