To find the total area of quadrilateral $ABCD$ in the image, we can split it into two triangles, $\triangle ABC$ and $\triangle ACD$, using diagonal $AC$ or $BD$. We can then find the area of each triangle individually and add them together.

Given Information:

• $AB = 65.8 \, \text{meters}$
• $BC = 66.6 \, \text{meters}$
• $CD = 32.8 \, \text{meters}$
• $DA = 64.4 \, \text{meters}$
• $AC = 94.1 \, \text{meters}$
• $BD = 102.6 \, \text{meters}$

Solution Steps:

1. Divide the Quadrilateral: We can divide quadrilateral $ABCD$ into two triangles by diagonal $AC$: $\triangle ABC$ and $\triangle ACD$.

2. Area of $\triangle ABC$: We know the side lengths $AB$, $BC$, and $AC$. We can use Heron's formula to find the area of $\triangle ABC$.

   • Calculate Semi-Perimeter $s$ for $\triangle ABC$: $s = \frac{AB + BC + AC}{2} = \frac{65.8 + 66.6 + 94.1}{2} = 113.25 \, \text{meters}$
   • Apply Heron's Formula: $\text{Area}_{ABC} = \sqrt{s \cdot (s - AB) \cdot (s - BC) \cdot (s - AC)}$ $= \sqrt{113.25 \cdot (113.25 - 65.8) \cdot (113.25 - 66.6) \cdot (113.25 - 94.1)}$

3. Area of $\triangle ACD$: Similarly, we use side lengths $AD$, $CD$, and $AC$ with Heron's formula.

   • Calculate Semi-Perimeter $s$ for $\triangle ACD$: $s = \frac{AD + CD + AC}{2} = \frac{64.4 + 32.8 + 94.1}{2} = 95.65 \, \text{meters}$
   • Apply Heron's Formula: $\text{Area}_{ACD} = \sqrt{s \cdot (s - AD) \cdot (s - CD) \cdot (s - AC)}$ $= \sqrt{95.65 \cdot (95.65 - 64.4) \cdot (95.65 - 32.8) \cdot (95.65 - 94.1)}$

4. Total Area of Quadrilateral $ABCD$: $\text{Total Area} = \text{Area}_{ABC} + \text{Area}_{ACD}$

Let me calculate each part.The areas of the triangles and the total area of quadrilateral $ABCD$ are as follows:

1. Area of $\triangle ABC$: $2191.03 \, \text{square meters}$
2. Area of $\triangle ACD$: $539.62 \, \text{square meters}$
3. Total Area of $ABCD$: $2730.64 \, \text{square meters}$

Would you like a breakdown of the calculations, or do you have any other questions?

Related Questions:

1. How would the area change if diagonal $BD$ was used to divide $ABCD$?
2. What other methods, besides Heron's formula, could we use to find the area of a quadrilateral?
3. How can we verify the accuracy of the measurements for complex shapes?
4. What are the conditions for using Heron's formula effectively?
5. Could we apply the cosine rule to determine any angles in this quadrilateral?

Tip:

When calculating areas of polygons, dividing them into triangles is a helpful strategy, especially when side lengths or diagonals are given.