1. **State the problem:** We have a triangle with vertices A(1,1), B(4,1), and C(1,5). We want to find the area of the triangle after it is dilated by a scale factor of $\frac{1}{2}$. 2. **Find the original area:** The triangle is right-angled with base AB along the x-axis and height AC along the y-axis. - Length of base $AB = 4 - 1 = 3$ - Length of height $AC = 5 - 1 = 4$ The area of a triangle is given by: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ So, $$\text{Area}_{original} = \frac{1}{2} \times 3 \times 4 = 6$$ 3. **Effect of dilation on area:** When a figure is dilated by a scale factor $k$, the area is multiplied by $k^2$. Here, $k = \frac{1}{2}$, so $$\text{Area}_{new} = \left(\frac{1}{2}\right)^2 \times \text{Area}_{original} = \frac{1}{4} \times 6 = \frac{6}{4}$$ 4. **Simplify the fraction:** $$\frac{6}{4} = \frac{\cancel{6}}{\cancel{4}} = \frac{3}{2} = 1.5$$ **Final answer:** The area of the resulting triangle after dilation is $1.5$ square units.