[Maths Class Notes] on Orthocenter Pdf for Exam

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)

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)

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)