1. **Problem statement:** Calculate the position of the centre of gravity (centroid) of the given composite shape with respect to point P, which is at the intersection of the X-X and Y-Y axes. 2. **Shape description and decomposition:** The shape consists of a rectangle and a right triangle on top to the right. - Rectangle dimensions: width $= 4$ m (2 + 1 + 1), height $= 2$ m. - Right triangle dimensions: base $= 1$ m (rightmost segment), height $= 3$ m. 3. **Coordinate system:** Point P is at the bottom-left corner of the rectangle. 4. **Step 1: Calculate area of each shape.** - Rectangle area $A_1 = 4 \times 2 = 8$ m$^2$. - Triangle area $A_2 = \frac{1 \times 3}{2} = 1.5$ m$^2$. 5. **Step 2: Find centroid coordinates of each shape relative to P.** - Rectangle centroid $(\bar{x}_1, \bar{y}_1)$ is at its center: $$\bar{x}_1 = \frac{4}{2} = 2, \quad \bar{y}_1 = \frac{2}{2} = 1$$ - Triangle centroid $(\bar{x}_2, \bar{y}_2)$ is located at $\frac{1}{3}$ from the base along the base and $\frac{1}{3}$ from the right angle along the height: The triangle is on the rightmost 1 m segment, so its base starts at $x=3$ m. $$\bar{x}_2 = 3 + \frac{1}{3} = 3.333...$$ $$\bar{y}_2 = 2 + \frac{3}{3} = 3$$ 6. **Step 3: Calculate the combined centroid coordinates $(\bar{x}, \bar{y})$ using the formula:** $$\bar{x} = \frac{A_1 \bar{x}_1 + A_2 \bar{x}_2}{A_1 + A_2}$$ $$\bar{y} = \frac{A_1 \bar{y}_1 + A_2 \bar{y}_2}{A_1 + A_2}$$ 7. **Step 4: Substitute values and compute:** $$\bar{x} = \frac{8 \times 2 + 1.5 \times 3.333...}{8 + 1.5} = \frac{16 + 5}{9.5} = \frac{21}{9.5} = 2.21 \text{ m}$$ $$\bar{y} = \frac{8 \times 1 + 1.5 \times 3}{9.5} = \frac{8 + 4.5}{9.5} = \frac{12.5}{9.5} = 1.32 \text{ m}$$ 8. **Final answer:** The centre of gravity is located approximately at $$\boxed{(\bar{x}, \bar{y}) = (2.21, 1.32) \text{ meters from point P}}$$