1. **Stating the problem:** Determine the area of the polygon in part (a) with sides 5.3 cm, 3 cm, 3.5 cm, 6.1 cm, and 2.5 cm. 2. **Understanding the problem:** We are given a polygon with various side lengths. To find the area, we can try to divide it into simpler shapes such as triangles or rectangles, then sum their areas. 3. **Approach:** Since the polygon is irregular, we can use the formula for the area of a triangle if we can identify triangles inside or use coordinate geometry if coordinates are known. Here, we assume the polygon can be split into triangles. 4. **Using Heron's formula for triangles:** Heron's formula for the area of a triangle with sides $a$, $b$, and $c$ is: $$ A = \sqrt{s(s-a)(s-b)(s-c)} $$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter. 5. **Applying to part (a):** Assuming the polygon can be split into two triangles with sides: - Triangle 1: sides 5.3 cm, 3 cm, 6.1 cm - Triangle 2: sides 3.5 cm, 2.5 cm, 6.1 cm Calculate semi-perimeters: $$ s_1 = \frac{5.3 + 3 + 6.1}{2} = \frac{14.4}{2} = 7.2$$ $$ s_2 = \frac{3.5 + 2.5 + 6.1}{2} = \frac{12.1}{2} = 6.05$$ Calculate areas: $$ A_1 = \sqrt{7.2(7.2-5.3)(7.2-3)(7.2-6.1)} = \sqrt{7.2 \times 1.9 \times 4.2 \times 1.1} = \sqrt{62.9856} \approx 7.94\,\text{cm}^2 $$ $$ A_2 = \sqrt{6.05(6.05-3.5)(6.05-2.5)(6.05-6.1)} = \sqrt{6.05 \times 2.55 \times 3.55 \times \cancel{-0.05}} $$ Since one term is negative, this triangle is not valid with these sides, so the polygon division assumption is incorrect. 6. **Alternative approach:** Without coordinates or angles, exact area calculation is not possible from side lengths alone. **Final answer:** Cannot determine the exact area of polygon (a) with given data alone.