Plane geometry for surveying computations: Using only the base AC and its corresponding perpendicular height h, what is the correct expression for the area of triangle ABC?

Introduction / Context:
Area relations are frequently needed in surveying for lot calculations, offsets, and coordinate methods. When a base and its perpendicular height are known, the area of a triangle reduces to a simple, universally known formula. The question deliberately restricts you to base AC and the perpendicular height h to ensure a unique correct expression.

Given Data / Assumptions:

• AC is a known side of triangle ABC, chosen as the base.
• h is the perpendicular distance from the opposite vertex to base AC.
• No other sides or angles are required for this specific method.

Concept / Approach:
For any triangle, area = (1/2) * base * height when the height is perpendicular to the chosen base. Although there are other correct formulae (sine rule variants and Heron’s formula), this problem requires the version involving base AC and its perpendicular height only, leading uniquely to 0.5 * AC * h.

Step-by-Step Solution:

Verification / Alternative check:
If other data were given, equivalent forms exist: Area = 0.5 * b * c * sin A, etc., or Heron’s formula using the semiperimeter S. However, because the problem restricts the inputs to AC and h only, those forms are intentionally out of scope.

Why Other Options Are Wrong:

• AC * h: missing the 0.5 factor, so it doubles the true area.
• 0.5 * a * c * sin B: correct in general but requires sides a, c and angle B, which are not part of the stated method.
• sqrt( S * (S - a) * (S - b) * (S - c) ): Heron’s formula is correct but does not use AC and h as requested.

Common Pitfalls:
Using the wrong height (oblique distance rather than perpendicular); forgetting the 0.5 factor; mixing base symbols from different conventions.

Final Answer:
Area = 0.5 * AC * h