Geometric Setup

The problem involves a surveyor measuring the distance to a reference point C. The surveyor's position is denoted as 'O'. A horizontal line segment AB, 200 m long, is centered at 'O'. This means OA = OB = 100 m.

Angles <CAB> and <CBA> are measured as 87.8 degrees. These angles define a triangle ABC.

Since <CAB> = <CBA>, triangle ABC is an isosceles triangle, with sides AC = BC.

In an isosceles triangle, the line segment from the apex (C) to the midpoint of the base (O) is perpendicular to the base. Thus, CO is perpendicular to AB, making <AOC> = 90 degrees.

Trigonometric Calculation

The goal is to find the distance OC, which is the distance from the surveyor's position ('O') to point C.

Focus on the right-angled triangle AOC:

• The length of the adjacent side AO is 100 m.
• The angle <CAO> is 87.8 degrees.
• The length of the opposite side OC needs to be calculated.

Using the tangent trigonometric relationship:

$ \tan(\angle CAO) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{OC}{AO} $

Substitute the known values into the equation:

$ \tan(87.8^\circ) = \frac{OC}{100 \text{ m}} $

Rearrange the formula to solve for OC:

$ OC = 100 \text{ m} \times \tan(87.8^\circ) $

Calculate the tangent value:

$ \tan(87.8^\circ) \approx 26.0306 $

Compute the distance OC:

$ OC \approx 100 \text{ m} \times 26.0306 $

$ OC \approx 2603.06 \text{ m} $

The distance to the reference point C is approximately 2603 m when rounded to the nearest meter.