1. **State the problem:** We need to find the angle $a$ at vertex $C$ in a scalene triangle $ABC$ drawn on a unit grid, where the side lengths are not aligned to the grid. 2. **Identify coordinates:** Assume the coordinates of points based on the unit grid: - Let $A = (x_A, y_A)$ - Let $B = (x_B, y_B)$ - Let $C = (x_C, y_C)$ 3. **Calculate vectors:** The angle $a$ at $C$ is between vectors $\overrightarrow{CA}$ and $\overrightarrow{CB}$: $$\overrightarrow{CA} = (x_A - x_C, y_A - y_C)$$ $$\overrightarrow{CB} = (x_B - x_C, y_B - y_C)$$ 4. **Use the dot product formula:** The angle between two vectors $\mathbf{u}$ and $\mathbf{v}$ is given by: $$\cos a = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}$$ where $$\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y$$ and $$|\mathbf{u}| = \sqrt{u_x^2 + u_y^2}$$ 5. **Calculate dot product and magnitudes:** Calculate $\overrightarrow{CA} \cdot \overrightarrow{CB}$ and the magnitudes $|\overrightarrow{CA}|$ and $|\overrightarrow{CB}|$. 6. **Calculate angle $a$:** $$a = \arccos\left(\frac{\overrightarrow{CA} \cdot \overrightarrow{CB}}{|\overrightarrow{CA}| |\overrightarrow{CB}|}\right)$$ Convert the result from radians to degrees. 7. **Example with coordinates:** Suppose from the grid: - $A = (0, 2)$ - $B = (3, 0)$ - $C = (1, 3)$ Calculate vectors: $$\overrightarrow{CA} = (0 - 1, 2 - 3) = (-1, -1)$$ $$\overrightarrow{CB} = (3 - 1, 0 - 3) = (2, -3)$$ Dot product: $$\overrightarrow{CA} \cdot \overrightarrow{CB} = (-1)(2) + (-1)(-3) = -2 + 3 = 1$$ Magnitudes: $$|\overrightarrow{CA}| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$$ $$|\overrightarrow{CB}| = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}$$ Calculate cosine: $$\cos a = \frac{1}{\sqrt{2} \times \sqrt{13}} = \frac{1}{\sqrt{26}}$$ Calculate angle $a$ in degrees: $$a = \arccos\left(\frac{1}{\sqrt{26}}\right) \approx \arccos(0.1961) \approx 78.7^\circ$$ **Final answer:** $$a \approx 78.7^\circ$$ This is the angle at vertex $C$ correct to 1 decimal place.