Orthocenter

Orthocentre of a Triangle Properties

Consider a triangle ABC in which the altitudes are drawn from the vertex to the opposite side of the vertex such that it forms a right angle with the side. Triangles have three vertices so these three altitudes are drawn will intersect at a certain point and that point is said to be the orthocenter of the respective triangle.

For example:

Consider a triangle ABC

Image


Here AD, BE and CF are the altitudes drawn on the sides BC, AC and AB respectively, all these three altitudes intersect at a point O. And so we can say that O is the orthocentre of a triangle ABC.

Orthocenter Definition

We can define orthocenter as:

In any given triangle the point of intersection of altitudes that are drawn perpendicular from the vertex to the opposite sides is called the Orthocenter of a triangle.

Orthocenter Properties

The orthocenter properties of a triangle depend on the type of a triangle.

The orthocentre of triangle properties are as follows:

  1. If a given triangle is the Acute triangle the orthocenter lies inside the triangle.

  2. If a given triangle is the Obtuse triangle the orthocenter lies outside the triangle.

  3. If a given triangle is the right-angled triangle the orthocenter lies on the triangle.

How to Calculate Orthocenter of a Triangle :

The orthocenter of a triangle can be calculated as follows:

Step 1: Let us calculate the slopes of the sides of the given triangle. 

  To calculate the slope we have,

Slope of a line= (y2 - y1)/(x2- x1).

Where ( x1 y1) are the x coordinates

and (x2 y2) are the y-coordinates

Step 2: Then we have to calculate the slopes of altitudes of the triangle. Altitudes are the perpendicular drawn from the vertex to the sides.

 To calculate the perpendicular slope we have,

Perpendicular Slope of Line = - (1/slope of a line)

Step 3: Then by using the point-slope form, calculate the equation for the altitudes with their respective coordinates. The point-slope formula is given as,

Point Slope Formula, (y - y1)=m(x - x1)

Step 4: Finally by solving any two altitude equations, we can get the orthocenter of the triangle.

Solved Examples:

Example: Find the Orthocenter of the Triangle with the Given Vertices: 

X(5, 3), Y(3, -1), Z(4, 2)

Solution:-

To find the orthocenter of triangle XYZ,

X(5, 3), Y(3, -1), Z(4, 2)


O is the Orthocenter of altitudes drawn from X, Y and Z.

The slope of XY with X ( 5, 3) and Y(3, -1)

Slope of a Line= (y2 - y1)/(x2- x1).

   =(-1 -3)/(3 -5)

       = --4/-2 =2

To calculate the perpendicular slope we have,

Perpendicular Slope of Line = - (1/slope of a line)

           Equation of altitude through Z(4, 2) is perpendicular to  XY,

           Perpendicular slope  = - ½


Then by using the point-slope form, calculate the equation for the altitudes with their respective coordinates. The point-slope formula is given as,

            Point Slope Formula,

             (y - y1)=m(x - x1)

  • With point z(4, 2) 

  • (y -2) = -½ (x - 4)

  • y - 2 = - ½ x  + 2 

  • ½ x + y  = 4……..(1)

Now, the slope of side YZ with Y( 3, -1) and Z(4, 2)

Slope of YZ = (2+1)/(4-3)

=3/1  =3

Perpendicular slope of YZ = -1/3 

Point slope formula, (y - y1)=m(x - x1)

  • (y-3) = -1/3( x -5)

  • y - 3 = -1/3x + 5/3

  • 1/3x + y = 5/3 +3

  • 1/3x + y = 14/3………………(2)

Solving equation 1 and 2 we get, the values of

x = -4 and y = 10/3

thus , we get the coordinates of Orthocenter as ( -4 , 10/3)

Example 2: If the Coordinates of the Vertices of Triangle ABC are A(0,0
), B( 3,0) and C(0,4) then Find the Orthocenter of the Triangle.

Solution:

The vertices of the triangle are A(0,0), B( 3,0) and C( 0,4). Here you can see we have AB on the Y- axis and AC passes through point zero, which shows that triangle is a right angled triangle.

Therefore, orthocenter lies on the triangle I.e Orthocenter is ( 0,0)

Quiz Time

1. Find the orthocenter of the triangle with the given vertices: 

P(2, 1), Q(5, 3), R(4, -1)


2. Find the orthocenter of the triangle with the given vertices: 

X(3, 5), Y(3, -4), Z(2, 3)

FAQ (Frequently Asked Questions)

1. What is Orthocenter?

Answer: in a triangle a point of intersection of all the three altitudes is said to be orthocenter. An altitude of a triangle is a line passing through the vertex of a triangle such that it is perpendicular to the opposite side of the vertex. Since the triangle has three vertices, we have three altitudes in the triangle. The orthocenter of a triangle varies according to the triangles. Different triangles like an equilateral triangle, isosceles triangle, scalene triangle, etc will have different altitudes

Consider the figure, Image


In triangle ABC AD, BE, CF are the altitudes drawn on the sides BC, AC and AB respectively. These altitudes intersect each other at point O. And this point O is said to be the orthocenter of the triangle ABC.

2. What is the Use of Orthocenter?

Answer: The Orthocenter of a triangle is used to identify the type of a triangle.

When the position of an Orthocenter of a triangle is given,

  1. If the Orthocenter of a triangle lies in the center of a triangle then the triangle is an acute triangle.

  2. If the Orthocenter of a triangle lies outside the triangle then the triangle is an obtuse triangle.

  3. If the Orthocenter of a triangle lies on the triangle then the triangle is a right-angled triangle.

  4. Besides this, the Orthocenter has several other properties related to circumcenter, incenter, and area of a triangle.