1. **Problem 8a:** Find the number of sides of a regular polygon with an internal angle of 150°. The formula for the internal angle $I$ of a regular polygon with $n$ sides is: $$I = \frac{(n-2) \times 180}{n}$$ We know $I = 150$, so: $$150 = \frac{(n-2) \times 180}{n}$$ 2. Multiply both sides by $n$: $$150n = 180(n-2)$$ 3. Expand the right side: $$150n = 180n - 360$$ 4. Rearrange to isolate $n$: $$180n - 150n = 360$$ $$30n = 360$$ 5. Divide both sides by 30: $$n = \frac{360}{30} = 12$$ **Answer:** The polygon has 12 sides. 6. **Problem 8b:** Find the perimeter of the polygon given each side length is 10 cm. The perimeter $P$ of a polygon is: $$P = n \times \text{side length}$$ 7. Substitute $n=12$ and side length = 10 cm: $$P = 12 \times 10 = 120$$ **Answer:** The perimeter is 120 cm. 8. **Problem 9:** Determine if point $(3,1)$ lies within the region defined by: $$3x + 2y \geq 12$$ $$5x + 3y \leq 15$$ 9. Substitute $x=3$, $y=1$ into the first inequality: $$3(3) + 2(1) = 9 + 2 = 11$$ Check if $11 \geq 12$? No, it is false. 10. Since the first inequality is false, the point $(3,1)$ does not satisfy the region's conditions. **Answer:** The point $(3,1)$ does not lie within the region $R$.