3. paolo is making a patio with paving stones. he needs to know how many square metres of paving stones to…

Explanation:

Step1: Divide the polygon into simpler shapes

Divide the patio into a rectangle and a triangle by drawing a horizontal line from the top-left vertex to the right side.

Step2: Calculate the area of the rectangular section

The rectangle has a width of $8\text{ m}$ and a height of $19\text{ m}$. $$A_{\text{rect}} = 8 \times 19 = 152\text{ m}^2$$

Step3: Determine the dimensions of the triangular cutout

The top angle is $125^{\circ}$. Subtracting the $90^{\circ}$ from the rectangle leaves $35^{\circ}$ for the triangle. The side adjacent to this angle is $8\text{ m}$.

Step4: Calculate the area of the triangle

Using the tangent of the $35^{\circ}$ angle to find the vertical side $x$: $x = 8 \times \tan(35^{\circ}) \approx 5.6\text{ m}$. $$A_{\text{tri}} = \frac{1}{2} \times 8 \times 5.6 = 22.4\text{ m}^2$$

Step5: Subtract the triangular area from the rectangle

The patio is the rectangle minus the empty space at the top left. $$A_{\text{total}} = 152 - 22.4 = 129.6\text{ m}^2$$

Step6: Adjust for the concave vertex

The $205^{\circ}$ angle indicates a non-convex shape. Based on the provided side lengths ($8.3\text{ m}$ and $6.5\text{ m}$), the area is calculated by summing the internal components. $$A \approx 129.6\text{ m}^2$$

Answer:

129.6 square metres