In given triangle point of intersection of altitudes that are drawn perpendicular from vertex to the opposite side is called as orthocentre of a triangle. The orthocenter has a relationship with other parts of the circle like the area, circumcenter, incenter, etc. An orthocenter can be used to find the type of triangle. There is no exact formula to calculate the orthocenter of a triangle. In this article, we find out how to calculate a triangle’s orthocenter.
Orthocenter definition
It is a point where all three altitudes of a triangle intersect each other.
Since a triangle has three vertices and three sides and therefore this has three altitudes.
Orthocentre varies for different triangles like equilateral, right-angled triangles in some triangles the position will be different.
It does not necessarily lie inside the triangle but it also can be outside the triangle e.g. obtuse triangle.
What are the properties of orthocentre?
It varies according to the type of triangle.
- If the triangle is an acute triangle then the orthocentre lies inside the triangle.
- If the triangle is an obtuse triangle then the orthocentre lies outside the triangle.
- If the triangle is a right-angle triangle then the orthocentre lies on the triangle.
How to calculate the orthocenter of a triangle?
Step 1: Calculate the slopes of the sides of a given triangle:
To calculate the slope,
Slope of a line =(y2-y1)/(x2-x1).
Where,
(x1 y1) are x coordinates
(x2 y2) are y co ordinates
Step 2: Now calculate the slope of altitudes of the triangle, altitudes are perpendicular drawn vertex to side.
For calculating perpendicular slope:
The perpendicular slope of line= -(1/ slope of the line)
Step 3: using point-slope form calculate the equation for altitude with respective altitudes.
The point slope formula is given as:
Point slope formula, (y-y1)=m(x- x1)
Step 4: solving any two altitudes to get orthocenter(x,y).
Construction of orthocenter
There is no specific formula to construct the triangle, but we can follow the basic steps to get the orthocenter of a triangle. Each triangle will have a unique orthocenter, so it is difficult to predict by any formula.
Step 1: Draw the altitudes from each of the three vertices to the opposite sides. If the altitudes do not fall on the sides then extend the sides(like in the case of the obtuse-angled triangle).
Step 2: The point at which the lines intersect is the orthocenter. In, the case if they do not intersect then follow step 3.
Step 3: Extend the altitudes in both directions, and you will get a point of intersection. This is the orthocenter, in this case, it lies outside the triangle.
Orthocenter of an acute angled triangle
Step 1: Draw altitudes from the vertex to the opposite sides. You will have three altitudes for three corresponding vertices.
Step 2: For an acute angled triangle all three vertices will intersect at a particular point, which is called the orthocenter of the triangle.
If in case the altitudes do not intersect then check the angle of the altitudes to be exactly 90 degrees.
Orthocenter of an obtuse angled triangle
Step 1: Draw the altitudes
Step 2: The altitudes do not intersect inside the circle.
Step 3: Extend the altitudes to find the point of the intersection outside the triangle.
Orthocenter of a right angled triangle
Step 1: Draw the altitudes. The two sides of the triangle are already altitudes.
Step 2: The altitudes intersect on the vertex opposite to the hypotenuse of the triangle.
Examples of how to find orthocenter of a triangle
Example: Consider a triangle ABC. If coordinates of vertices of triangle ABC are A (0,0), B(30,0), C(0,50). Find the orthocentre.
Follow the steps mentioned above to calculate the triangle orthocentre.
Draw the triangle according to the coordinates and draw the altitudes. We find that it is a right-angled triangle. A right-angled triangle has its orthocenter at the right-angled vertex. And hence, the orthocentre lies on the triangle at (0,0).
Solve a question
If the given vertices are P(5,1), Q(0,3), R(10,14). Find the orthocenter of this triangle.
See also
Area of a sector formula
Perimeter of a sector-Formula
Perimeter of right-angled triangle
Every square is a rectangle
Volume of a parallelepiped
