A farm is in the shape of a trapezium ABCD, with parallel sides AD and BC. Describe how to find the area of the farm using measurements taken from a car.

The assessor drives the car along each side of the farm (AB, BC, CD, and DA) and records the distance traveled for each segment using the car's odometer. Let these lengths be $L_{AB}$, $L_{BC}$, $L_{CD}$, and $L_{DA}$.

The assessor drives the car across the farm along one of its diagonals, for example, from point A to point C, and records this distance using the odometer. Let this length be $L_{AC}$.

The diagonal $L_{AC}$ divides the farm (quadrilateral ABCD) into two triangles: $\triangle ABC$ and $\triangle ADC$.

Use Heron's formula for each triangle.

• For $\triangle ABC$ (sides $L_{AB}$, $L_{BC}$, $L_{AC}$):
  • Calculate the semi-perimeter $s_1 = \frac{L_{AB} + L_{BC} + L_{AC}}{2}$.
  • Area($\triangle ABC$) $= \sqrt{s_1(s_1 - L_{AB})(s_1 - L_{BC})(s_1 - L_{AC})}$.
• For $\triangle ADC$ (sides $L_{AD}$, $L_{DC}$, $L_{AC}$):
  • Calculate the semi-perimeter $s_2 = \frac{L_{AD} + L_{DC} + L_{AC}}{2}$.
  • Area($\triangle ADC$) $= \sqrt{s_2(s_2 - L_{AD})(s_2 - L_{DC})(s_2 - L_{AC})}$.

The total area of the farm is the sum of the areas of the two triangles.

Method 2: Using the car's GPS system (if available)

If the car is equipped with a GPS system, the assessor can:

Drive to each vertex of the farm (A, B, C, D) and record its geographical coordinates (latitude and longitude) using the GPS.

Use these coordinates with a suitable geodetic area calculation method, such as the Shoelace formula (also known as Gauss's area formula), which can compute the area of a polygon given the coordinates of its vertices.