To find the orthocenter of a triangle, we follow these steps:

1. Understand the orthocenter: It is the intersection of the altitudes of a triangle. An altitude is a line passing through a vertex and perpendicular to the opposite side.

2. Given triangle vertices:
   $A(-1, 4)$, $B(7, -4)$, $C(7, 4)$.

3. Step 1 - Find the slopes of sides $AB$ and $AC$:

   • Slope of $AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 4}{7 - (-1)} = \frac{-8}{8} = -1$.
   • Slope of $AC = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 4}{7 - (-1)} = \frac{0}{8} = 0$.

4. Step 2 - Find the slopes of the altitudes:
   Altitudes are perpendicular to the sides, so their slopes are negative reciprocals:

   • Altitude from $C$ to $AB$: Slope = $1$ (reciprocal of $-1$).
   • Altitude from $B$ to $AC$: Slope = Undefined (since $AC$ is horizontal, its altitude is vertical).

5. Step 3 - Equations of the altitudes:

   • Altitude from $C(7, 4)$: Passes through $C$ with slope $1$.
     Using the point-slope form $y - y_1 = m(x - x_1)$:
     $y - 4 = 1(x - 7) \implies y = x - 3.$
   • Altitude from $B(7, -4)$: Vertical line through $B$:
     $x = 7.$

6. Step 4 - Find the intersection of the altitudes:
   Solve the system of equations: $x = 7, \quad y = x - 3.$ Substituting $x = 7$ into $y = x - 3$:
   $y = 7 - 3 = 4.$

Thus, the orthocenter is $(7, 4)$.

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Do you want further clarification or details? Here are 5 questions to explore further:

1. How is the orthocenter different from other triangle centers like the centroid or circumcenter?
2. Can the orthocenter lie outside the triangle? Under what conditions?
3. How would the orthocenter be calculated in a 3D triangle?
4. What is the significance of perpendicularity in the calculation of altitudes?
5. How does the orthocenter's location relate to right triangles?

Tip: For a right triangle, the orthocenter lies at the vertex of the right angle!