To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.

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Identify the triangle involved: points A, B, and C form a triangle where BC = 354 m is known, and angles at B and C are given as 112° 10' and 15° 20' respectively.

Convert the given angles from degrees and minutes to decimal degrees for easier calculation: for example, 112° 10' = 112 + 10/60 degrees.

Calculate the third angle at point A using the triangle angle sum property: \(A = 180^\circ - B - C\).

Use the Law of Sines to relate the sides and angles in the triangle: \(\frac{AB}{\sin C} = \frac{BC}{\sin A}\).

Rearrange the Law of Sines formula to solve for AB: \(AB = \frac{BC \times \sin C}{\sin A}\). Substitute the known values of BC, \(\sin C\), and \(\sin A\) to find AB.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Law of Sines

The Law of Sines relates the sides and angles of any triangle, stating that the ratio of a side length to the sine of its opposite angle is constant. It is essential for solving triangles when two angles and one side or two sides and one angle are known, enabling calculation of unknown sides or angles.

Angle Measurement and Conversion

Understanding how to interpret and convert angles given in degrees and minutes is crucial. Since angles are often provided in degrees (°) and minutes ('), converting these to decimal degrees or radians facilitates accurate calculations in trigonometric formulas.

Triangle Geometry in Surveying

Surveying problems often involve forming triangles between points on the ground and using measured distances and angles to find unknown lengths. Recognizing how to apply trigonometric principles to real-world layouts, such as across a river, is key to solving for distances like AB.

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