1. **State the problem:** Calculate the area of the triangle with vertices at points $A(1,2)$, $B(9,2)$, and $C(6,7)$. 2. **Formula for the area of a triangle:** The area $A$ of a triangle is given by $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Identify the base and height:** The base is the segment $AB$ along the horizontal line $y=2$. The length of the base is the distance between points $A$ and $B$: $$\text{base} = 9 - 1 = 8$$ 4. **Calculate the height:** The height is the vertical distance from point $C$ to the base line $y=2$. Since $C$ has $y$-coordinate 7, the height is: $$\text{height} = 7 - 2 = 5$$ 5. **Calculate the area:** Substitute the base and height into the formula: $$A = \frac{1}{2} \times 8 \times 5 = \frac{1}{2} \times 40 = 20$$ 6. **Final answer:** The area of the triangle is $20$ square units.