1. **State the problem:** Find the perimeter and area of the triangle with vertices at points (4,1), (6,1), and (5,8) on the coordinate plane. 2. **Formula for perimeter:** The perimeter $P$ is the sum of the lengths of all sides. Use the distance formula between two points $A(x_1,y_1)$ and $B(x_2,y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate side lengths:** - Side between (4,1) and (6,1): $$d_1 = \sqrt{(6-4)^2 + (1-1)^2} = \sqrt{2^2 + 0^2} = \sqrt{4} = 2$$ - Side between (6,1) and (5,8): $$d_2 = \sqrt{(5-6)^2 + (8-1)^2} = \sqrt{(-1)^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \approx 7.07$$ - Side between (5,8) and (4,1): $$d_3 = \sqrt{(4-5)^2 + (1-8)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \approx 7.07$$ 4. **Calculate perimeter:** $$P = d_1 + d_2 + d_3 = 2 + 7.07 + 7.07 = 16.14$$ 5. **Formula for area:** Use the shoelace formula or the formula for area of triangle given coordinates: $$A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ 6. **Calculate area:** $$A = \frac{1}{2} |4(1 - 8) + 6(8 - 1) + 5(1 - 1)| = \frac{1}{2} |4(-7) + 6(7) + 5(0)| = \frac{1}{2} |-28 + 42 + 0| = \frac{1}{2} |14| = 7$$ 7. **Final answers rounded to nearest hundredth:** - Perimeter $P \approx 16.14$ units - Area $A = 7.00$ units$^2$