How to find the coordinates of the orthocenter?
Understand the Problem
The question is asking about the process to determine the coordinates of the orthocenter of a triangle, which involves knowledge of geometry and the properties of triangle centers.
Answer
The coordinates of the orthocenter are $(h_x, h_y)$.
Answer for screen readers
The coordinates of the orthocenter are $(h_x, h_y)$, where $h_x$ and $h_y$ are computed from solving the simultaneous equations of the altitudes.
Steps to Solve
- Identify the Vertices of the Triangle
First, you need to know the coordinates of the vertices of the triangle, which we can label as $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$.
- Find the Slopes of the Sides
Calculate the slopes of the sides of the triangle. The slope of line AB can be found using the formula:
$$ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} $$
Similarly, calculate the slopes of lines AC and BC:
$$ m_{AC} = \frac{y_3 - y_1}{x_3 - x_1} $$
$$ m_{BC} = \frac{y_3 - y_2}{x_3 - x_2} $$
- Determine the Slopes of the Altitudes
The slopes of the altitudes can be determined by taking the negative reciprocal of the corresponding sides:
For altitude from vertex C to line AB:
$$ m_{hC} = -\frac{1}{m_{AB}} $$
For altitude from vertex A to line BC:
$$ m_{hA} = -\frac{1}{m_{BC}} $$
- Find the Equations of the Altitudes
Using the point-slope form of the equation of a line, write the equations for the altitudes. For example, the equation of the altitude from C can be written as:
$$ y - y_3 = m_{hC}(x - x_3) $$
Repeat this process for the other altitude.
- Solve the System of Equations
Now you have two equations (one for each altitude). Solve these equations simultaneously to find the coordinates of the orthocenter $H(h_x, h_y)$.
You can express this system as:
$$ \begin{cases} y - y_3 = m_{hC}(x - x_3) \ y - y_1 = m_{hA}(x - x_1) \end{cases} $$
- Calculate the Coordinates of the Orthocenter
Using substitution or elimination methods, solve for $h_x$ and $h_y$ to determine the coordinates of the orthocenter.
The coordinates of the orthocenter are $(h_x, h_y)$, where $h_x$ and $h_y$ are computed from solving the simultaneous equations of the altitudes.
More Information
The orthocenter of a triangle is one of its several notable points, formed by the intersection of the three altitudes of the triangle. The orthocenter's position can vary depending on the type of triangle: it is inside for acute triangles, on the vertex for right triangles, and outside for obtuse triangles.
Tips
- Mixing up the slopes of the lines and their corresponding altitudes.
- Forgetting to take the negative reciprocal when calculating the slopes of the altitudes.
- Miscalculating the intersection of the lines due to algebraic simplification errors.
