1. **Problem Statement:** We have a large square plate of side 320 mm with a smaller rotated square cut out inside it. The smaller square has side 60 mm and is positioned such that one vertex is 80 mm from the top and left edges, and the opposite vertex is 60 mm from the bottom and left edges. The coordinate system is centered at M, the center of the large square. We need to find the quadrant where the centroid of the shaded area lies and determine its coordinates. 2. **Understanding the setup:** - The large square has side length $320$ mm, so its area is $$A_1 = 320 \times 320 = 102400 \text{ mm}^2.$$ Its centroid is at the center $M$, coordinates $(0,0)$. - The smaller square has side length $60$ mm, so its area is $$A_2 = 60 \times 60 = 3600 \text{ mm}^2.$$ 3. **Coordinates of the smaller square centroid:** - The smaller square is rotated and positioned inside the large square. Given the distances from edges, we find the centroid of the smaller square relative to $M$. - Since the large square center $M$ is at $(0,0)$, the edges are at $\pm 160$ mm in both $x$ and $y$ directions. - The smaller square vertex is 80 mm from top and left edges, so its position relative to $M$ is: - Left edge of large square: $x = -160$ - Vertex of small square: $x = -160 + 80 = -80$ - Similarly for $y$: top edge $y=160$, vertex $y=160 - 80 = 80$ - The opposite vertex is 60 mm from bottom and left edges: - Bottom edge $y = -160$ - Vertex $y = -160 + 60 = -100$ - Left edge $x = -160$ - Vertex $x = -160 + 60 = -100$ - The smaller square is rotated, but its centroid is at the midpoint of the diagonal connecting these two vertices: $$x_c = \frac{-80 + (-100)}{2} = -90$$ $$y_c = \frac{80 + (-100)}{2} = -10$$ 4. **Centroid of the shaded area:** - The shaded area is the large square minus the smaller square. - Using the formula for composite areas: $$x_{centroid} = \frac{A_1 x_1 - A_2 x_2}{A_1 - A_2}$$ $$y_{centroid} = \frac{A_1 y_1 - A_2 y_2}{A_1 - A_2}$$ where $(x_1,y_1) = (0,0)$ is centroid of large square, and $(x_2,y_2) = (-90,-10)$ is centroid of smaller square. - Calculate: $$x_{centroid} = \frac{102400 \times 0 - 3600 \times (-90)}{102400 - 3600} = \frac{324000}{98700} \approx 3.28$$ $$y_{centroid} = \frac{102400 \times 0 - 3600 \times (-10)}{102400 - 3600} = \frac{36000}{98700} \approx 0.36$$ 5. **Quadrant location:** - Since $x_{centroid} \approx 3.28 > 0$ and $y_{centroid} \approx 0.36 > 0$, the centroid lies in the **first quadrant**. **Final answer:** - The centroid of the shaded area is located in the **first quadrant**. - Its coordinates are approximately $$\boxed{(3.28, 0.36)} \text{ mm}.$$