Orthocenter

Orthocenter of the triangle is the point of the triangle where all the three altitudes of the triangle meet or intersect each other. You must have learned various terms in case of triangles, such as area, perimeter, centroid, etc. Area defines the space covered, perimeter defines the length of the outer line of triangles and centroid is the point where all the lines drawn from the vertex of the triangle meets. Also learn, Circumcenter of a Triangle.

The orthocenter will vary for different types of triangles such as Isosceles, Equilateral, Scalene, right-angled, etc. In the case of an equilateral triangle, the centroid will be the orthocenter. But in the case of other triangles, the position will be different. It is not necessary that orthocenter lies inside the triangle only, in case of an obtuse triangle, it lies outside of the triangle.

Orthocenter of a Triangle

The orthocenter of a triangle is the point where the perpendicular drawn from the vertex to the opposite sides of the triangle intersect each other. Take an example of a triangle ABC.

Orthocenter of a Triangle

In the above figure, you can see, the perpendicular AD, BE and CF drawn from vertex A, B and C to the opposite sides BC, AC and AB, respectively, intersect each other at a single point O. This point is the orthocenter of △ABC.

How to Construct Orthocenter?

To construct the orthocenter of a triangle, there is no particular formula but we have to get the coordinates of the vertices of the triangle. Suppose we have a triangle ABC and we need to find the orthocenter of it. Then follow the below-given steps;

  • The first thing we have to do is find the slope of the side BC, using the slope formula, which is, m = y2-y1/x2-x1
  • The slope of the line AD is the perpendicular slope of BC.
  • Now, from the point, A and slope of the line AD, write the straight-line equation using the slope-intercept formula which is;

    y2-y1 = m (x2-x1)

  • Again find the slope of side AC using slope formula.
  • The perpendicular slope of AC is the slope of the line BE.
  • Now, from the point, B and slope of the line BE, write the straight-line equation using the slope-intercept formula which is;

    y-y1 = m (x-x1)

  • Now, we have got two equations for straight lines which is AD and BE.
  • Extend both the lines to find the intersection point.
  • The point where AD and BE meets is the orthocenter.

Construction of Orthocenter

Note: If we are able to find the slopes of the two sides of the triangle then we can find the orthocenter and its not necessary to find the slope for the third side also.

Properties of Orthocenter

Orthocenter is the intersection point of the altitudes drawn from the vertices of the triangle to the opposite sides.

  • For an acute triangle, it lies inside the triangle.
  • For an obtuse triangle, it lies outside of the triangle.
  • For right-angled triangle, it lies on the triangle.
  • The product of the parts into which the orthocenter divides an altitude is the equivalent for all 3 perpendiculars.

Example of Orthocenter

Question: Find the orthocenter of a triangle whose vertices are A (-5, 3), B (1, 7), C (7, -5).

Solution: Let us solve the problem with the steps given in the above section;

1. Slope of the side AB = y2-y1/x2-x1 = 7-3/1+5=4/6=⅔

2. The perpendicular slope of AB = -3/2

3. With point C(7, -5) and slope of CF = -1, the equation of CF is y – y1 = m (x – x1) ( slope intercept form)

4. Substitute the values in the above formula.

(y + 5) = -1(x – 7)

y + 5 = -x + 7

x + y = 2 ………………………………….(1)

5. Slope of side BC = y2-y1/x2-x1 = (-5-7)/(7-1) = -12/6=-2

6. The perpendicular slope of BC = ½

7. Now, the equation of line AD is y – y1 = m (x – x1) ( slope intercept form)

(y-3) = ½(x+5)

Solving the equation we get,

x-2y = -11…………………………………………(2)

8. Now when we solve equations 1 and 2, we get the x and y values.

Which are, x = 13/3 and y = -7/3

Therefore(13/3,-7/3) are the coordinates of orthocenter of the triangle.

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