Find The Orthocentre Of The Triangle With The Vertices

Finding the orthocentre of a triangle is a fundamental concept in geometry that involves determining the point where the altitudes of the triangle intersect. The orthocentre plays a crucial role in various geometric constructions and calculations. In this article, we will explore how to find the orthocentre of a triangle with given vertices, using both analytical and geometrical methods.

Understanding the Orthocentre

The orthocentre of a triangle is the point where the three altitudes of the triangle intersect. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. The orthocentre is denoted by the letter ‘H’ and is an important point in the triangle, as it is the vertex of the triangle’s orthic triangle (the triangle formed by the feet of the altitudes).

Analytical Method

To find the orthocentre of a triangle with given vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), we can use the following steps:

1. Calculate the slopes of the lines passing through each side of the triangle:
– Slope of line passing through (x₁, y₁) and (xâ‚‚, yâ‚‚): m₁ = (yâ‚‚ – y₁) / (xâ‚‚ – x₁)
– Slope of line passing through (xâ‚‚, yâ‚‚) and (x₃, y₃): mâ‚‚ = (y₃ – yâ‚‚) / (x₃ – xâ‚‚)
– Slope of line passing through (x₃, y₃) and (x₁, y₁): m₃ = (y₁ – y₃) / (x₁ – x₃)

2. Calculate the perpendicular slopes for each side of the triangle:
– Perpendicular slope to m₁: m₁⊥ = -1 / m₁
– Perpendicular slope to mâ‚‚: m₂⊥ = -1 / mâ‚‚
– Perpendicular slope to m₃: m₃⊥ = -1 / m₃

3. Calculate the equations of the altitudes passing through each vertex:
– Equation of altitude passing through (x₁, y₁): y – y₁ = m₁⊥(x – x₁)
– Equation of altitude passing through (xâ‚‚, yâ‚‚): y – yâ‚‚ = m₂⊥(x – xâ‚‚)
– Equation of altitude passing through (x₃, y₃): y – y₃ = m₃⊥(x – x₃)

4. Solve the system of equations formed by the three altitude equations to find the coordinates of the orthocentre (H).

Geometrical Method

Another method to find the orthocentre of a triangle is to use the properties of orthocentres:

1. Construct the altitudes of the triangle:
– Draw a perpendicular line from each vertex to the opposite side, forming three altitudes.
– Label the points where the altitudes intersect the opposite sides as A’, B’, and C’, respectively.

2. Extend the altitudes to meet at a point H:
– The point where the altitudes intersect (A’, B’, C’) is the orthocentre of the triangle.

Example:

Let’s consider a triangle with vertices at (1, 3), (4, 2), and (5, 6). Using the analytical method, we can find the orthocentre as follows:

1. Calculate the slopes:
– m₁ = (2 – 3) / (4 – 1) = -1 / 3
– mâ‚‚ = (6 – 2) / (5 – 4) = 4
– m₃ = (3 – 6) / (1 – 5) = 3 / 4

2. Calculate perpendicular slopes:
– m₁⊥ = -1 / (-1 / 3) = 3
– m₂⊥ = -1 / 4
– m₃⊥ = -1 / (3 / 4) = -4 / 3

3. Calculate altitude equations:
– Altitude through (1, 3): y – 3 = 3(x – 1)
– Altitude through (4, 2): y – 2 = (-1 / 4)(x – 4)
– Altitude through (5, 6): y – 6 = (-4 / 3)(x – 5)

4. Solve the system of equations to find the orthocentre.

Finding the orthocentre of a triangle involves calculating the point where the altitudes of the triangle intersect. This can be done analytically by determining the equations of the altitudes or geometrically by constructing the altitudes and finding their intersection. Both methods provide a way to identify this important point in a triangle.