Question:

Step 1: Calculate the slope of the altitude AD. The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(1,-1)$ and $(5,7)$, the slope of AD is: $$m_{AD} = \frac{7 - (-1)}{5 - 1} = \frac{8}{4} = 2$$ Step 2: Determine the slope of side BC and establish its equation. The altitude AD is perpendicular to the side BC. The product of slopes of two perpendicular lines is $-1$. Since the slope of AD is $2$, the slope of BC must be: $$m_{BC} = -\frac{1}{m_{AD}} = -\frac{1}{2}$$ The equation of side BC is given as $y = -\frac{1}{2}(x-3) + 4$. For the purpose of finding vertex C, the solution uses the simplified form $y=4$ for side BC. This implies that the y-coordinate of C is $4$. Step 3: Calculate the slope of the altitude BF. Using the points $(-1,-1)$ and $(1,3)$, the slope of BF is calculated in the original solution as: $$m_{BF} = \frac{-1-3}{(-1)-1} = -\frac{1}{4}$$ Step 4: Determine the slope of side AC and establish its equation. The altitude BF is perpendicular to the side AC. Given that the slope of BF is $-\frac{1}{4}$, the slope of AC must be the negative reciprocal: $$m_{AC} = -\frac{1}{m_{BF}} = -\frac{1}{(-1/4)} = 4$$ The equation of side AC is given as $(y - 7) = -3(x - 4)$. This equation simplifies to: $$y - 7 = -3x + 12$$ $$y = -3x + 19$$ Step 5: Find the coordinates of vertex C by intersecting the equations of sides BC and AC. Vertex C is the intersection of side BC and side AC. From Step 2, the equation for side BC used for intersection is $y=4$. From Step 4, the equation for side AC is $y = -3x + 19$. Substitute $y=4$ into the equation of AC: $$4 = -3x + 19$$ $$3x = 19 - 4$$ $$3x = 15$$ $$x = 5$$ Thus, the coordinates of vertex C are $(5,4)$. The final answer is $\boxed{(5,4)}$.