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To solve the problem of finding the orthocenter, we first need to identify the vertices of the triangle from the coordinate plane in the image.

Here’s how to proceed:

1. **Identify the vertices**: From the grid, you can see that the triangle's vertices are at:
   - \( A(0, 0) \)
   - \( B(6, 0) \)
   - \( C(0, 6) \)
   
2. **Understand the orthocenter**: The orthocenter of a triangle is the point where all three altitudes intersect. An altitude is a line segment from a vertex perpendicular to the opposite side.

3. **Altitudes of the triangle**: 
   - The altitude from vertex \( A(0,0) \) is along the vertical side of the triangle (the y-axis), which means it is already on the y-axis.
   - The altitude from vertex \( B(6,0) \) would be perpendicular to the line between \( A \) and \( C \), which is vertical, so this altitude lies on the x-axis.
   - The third altitude is drawn from vertex \( C(0,6) \) and is perpendicular to the base of the triangle (the line from \( B \) to \( A \)).

4. **Finding the orthocenter**: In this specific triangle, which is a right triangle, the orthocenter lies at the right angle vertex. Since the right angle is at vertex \( A(0,0) \), the orthocenter is at this point.

So, the coordinates of the orthocenter are \( \boxed{(0,0)} \).

Would you like more details on this? Here are five follow-up questions for further exploration:

1. What is the difference between the orthocenter and the centroid of a triangle?
2. How do you find the orthocenter for an obtuse triangle?
3. What is the relationship between the orthocenter and other triangle centers (like the circumcenter or incenter)?
4. How do you find the equation of an altitude for non-right triangles?
5. Why is the orthocenter the same as the vertex for a right triangle?

**Tip**: The orthocenter is always inside the triangle for an acute triangle but lies outside for obtuse triangles.

Sergio is creating a pattern that he is going to use to cut out triangular pieces of wood. He draws this pattern on a coordinate plane. What are the coordinates of the orthocenter?

Learn how to find the orthocenter of a right triangle drawn on a coordinate plane. This step-by-step solution covers the geometric properties and theorems related to triangle centers, specifically focusing on the orthocenter in a right triangle. Suitable for high school students.

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