How to find orthocenter with coordinates?
Understand the Problem
The question is inquiring about the method to find the orthocenter of a triangle when its vertices are provided as coordinates. This typically involves using the properties of the triangle's altitudes and their intersection point.
Answer
The orthocenter is found by solving for the intersection of any two altitudes from the triangle's vertices.
Answer for screen readers
The orthocenter of the triangle with vertices ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ) can be found as the intersection point of the altitudes from at least two vertices.
Steps to Solve
- Identify the triangle vertices
Label the triangle vertices with their coordinates. Let the vertices be ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ).
- Calculate the slopes of the sides
Find the slopes of the sides of the triangle using the formula for slope given two points:
The slope ( m_{AB} ) between points ( A ) and ( B ) is given by:
$$ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} $$
Repeat for ( m_{BC} ) and ( m_{CA} ).
- Determine the slopes of the altitudes
The altitude from a vertex is perpendicular to the opposite side. The slope of the altitude from vertex ( A ), which is perpendicular to side ( BC ), is:
$$ m_{hA} = -\frac{1}{m_{BC}} $$
Similarly, calculate the slopes ( m_{hB} ) and ( m_{hC} ) for the altitudes from vertices ( B ) and ( C ).
- Find the equations of the altitudes
Using point-slope form, the equation of the altitude from vertex ( A ) can be written as:
$$ y - y_1 = m_{hA}(x - x_1) $$
Do this for altitudes from vertices ( B ) and ( C ).
- Solve for intersections of altitudes
You now have two equations (e.g., altitude from ( A ) and altitude from ( B )). Set these equations equal to each other to find their intersection point:
$$ y_{hA} = y_{hB} $$
This will give you the coordinates of the orthocenter.
- Conclusion
The coordinates obtained from the intersection of any two altitudes provide the coordinates of the orthocenter of the triangle.
The orthocenter of the triangle with vertices ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ) can be found as the intersection point of the altitudes from at least two vertices.
More Information
The orthocenter is one of the triangle's key centers, along with the centroid and circumcenter. Its position can vary depending on whether the triangle is acute, right, or obtuse. In an acute triangle, it lies inside the triangle; in a right triangle, it lies on the vertex of the right angle; and in an obtuse triangle, it lies outside.
Tips
- Not calculating the slopes correctly, leading to incorrect equations of the altitudes.
- Forgetting to check for vertical slopes, which require special handling since their slope is undefined.
- Mixing up the altitude equations or their corresponding vertices, causing errors in finding the intersection point.
