When the diagonals of a trapezium split it into four triangles, the total area can be found if you know the areas of the two parallel-base triangles ($A_1$ and $A_2$):
$\text{Total Area} = (\sqrt{A_1} + \sqrt{A_2})^2$
1. Identify the parallel-base areas
From the problem breakdown, the two triangles touching the parallel sides have areas of 45 and 5 sq. units.
2. Apply the formula
Substitute these values directly into the square-root formula:
$\text{Total Area} = (\sqrt{45} + \sqrt{5})^2$
3. Simplify the square roots
Simplify $\sqrt{45}$ by breaking it down into prime factors ($\sqrt{9 \times 5}$):
$\sqrt{45} = 3\sqrt{5}$
4. Combine like terms
Add the two terms inside the parentheses together:
$\text{Total Area} = (3\sqrt{5} + 1\sqrt{5})^2$
$\text{Total Area} = (4\sqrt{5})^2$
5. Final squaring
Square both parts of the term ($4^2$ and $(\sqrt{5})^2$):
$\text{Total Area} = 16 \times 5 = \mathbf{80\text{ sq. units}}$
