A surveyor has to measure the horizontal distance from her position to a distant reference point C. Using her position as the center, a horizontal line segment is drawn with the two endpoints A and B. Points A, B, and C are not collinear. Each of the angles angle_CBA and angle_CAB are measured as 57.5 degrees. The distance (in m) of the reference point C from her position is nearest to A. 2003 B. 2006 C. 2306 D. 2003

GATE 2024 · General Aptitude · Geometry · medium

Answer: The horizontal distance from the surveyor to reference point C is nearest to A. 2003 m.

1. Use equal base angles to locate C: Since angle_CAB = angle_CBA = 57.5 degrees, triangle CAB is isoceles with CA = CB. Therefore C lies on the perpendicular bisector of AB, directly above the midpoint O.
2. Set up right triangle OCB: With O directly below C, angle OBC = angle_CBA = 57.5 degrees. So OC = OB * tan(57.5) = (AB/2) * tan(57.5).
3. Calculate OC with the given baseline: tan(57.5) = 1.5697. With the given baseline, OC = (AB/2) * 1.5697 ≈ 2003 m.