1. **Problem Statement:** Find the centroid of the first composite L-shaped figure with given dimensions in mm: vertical height 60 mm, horizontal top width 80 mm, vertical lower inner gap -20 mm, total height 90 mm, and small vertical thickness 30 mm. 2. **Approach:** The centroid of a composite shape can be found by dividing it into simpler rectangles, finding each rectangle's centroid and area, then using the formula: $$\bar{x} = \frac{\sum A_i x_i}{\sum A_i}, \quad \bar{y} = \frac{\sum A_i y_i}{\sum A_i}$$ where $A_i$ is the area and $(x_i, y_i)$ is the centroid of each part. 3. **Divide the shape into two rectangles:** - Rectangle 1 (vertical part): width = 30 mm, height = 90 mm - Rectangle 2 (horizontal part): width = 80 mm, height = 30 mm 4. **Calculate areas:** - $A_1 = 30 \times 90 = 2700$ mm$^2$ - $A_2 = 80 \times 30 = 2400$ mm$^2$ 5. **Find centroids of each rectangle relative to the bottom-left corner of the shape:** - Rectangle 1 centroid: $x_1 = \frac{30}{2} = 15$ mm, $y_1 = \frac{90}{2} = 45$ mm - Rectangle 2 centroid: $x_2 = \frac{80}{2} = 40$ mm, $y_2 = 90 - \frac{30}{2} = 75$ mm 6. **Calculate composite centroid coordinates:** $$\bar{x} = \frac{2700 \times 15 + 2400 \times 40}{2700 + 2400} = \frac{40500 + 96000}{5100} = \frac{136500}{5100} = 26.76 \text{ mm}$$ $$\bar{y} = \frac{2700 \times 45 + 2400 \times 75}{5100} = \frac{121500 + 180000}{5100} = \frac{301500}{5100} = 59.12 \text{ mm}$$ 7. **Final answer:** The centroid of the composite L-shaped figure is at $$\boxed{(26.76, 59.12) \text{ mm}}$$