Orthocenter
Orthocenter indicates the center of all the right angles from the vertices to the opposite sides i.e., the altitudes. The term ortho means right and it is considered to be the intersection point of three altitudes drawn from the three vertices of a triangle. An orthocentre has significant importance in the study of the various properties of a triangle with respect to its other dimensions. Let us learn more about the orthocenter properties, orthocenter formula, orthocenter definition, and solve a few examples.
| 1. | Definition of an Orthocenter |
| 2. | Properties of an Orthocenter |
| 3. | How to Construct an Orthocenter? |
| 4. | Orthocentre of a Triangle Formula |
| 5. | FAQs on Orthocenter |
Definition of an Orthocenter
An orthocenter can be defined as the point of intersection of altitudes that are drawn perpendicular from the vertex to the opposite sides of a triangle. In a triangle, it is that point where all the three altitudes of a triangle intersect. The main three main aspects of an orthocenter are:
- Triangle - A polygon with three vertices and three edges.
- Altitude - The altitude of a triangle is that line that passes through its vertex and is perpendicular to the opposite side. Hence, a triangle can have three altitudes, one from each vertex.
- Vertex - The point where two or more lines meet is called a vertex.
Look at the image below, △ABC is a triangle, △ABC has three altitudes, namely, AE, BF, and, CD, △ABC has three vertices, namely, A, B, and, C, and the intersection point H is the orthocenter.
Properties of an Orthocenter
The properties of an orthocenter vary depending on the type of triangle such as the Isosceles triangle, Scalene triangle, right-angle triangle, etc. For some triangles, the orthocenter need not lie inside the triangle but can be placed outside. For instance, for an equilateral triangle, the orthocenter is the centroid. The properties are as follows:
Property 1: The orthocenter lies inside the triangle for an acute angle triangle. As seen in the below figure, the orthocenter is the intersection point of the lines PF, QS, and RJ.
Property 2: The orthocenter lies outside the triangle for an obtuse angle triangle. As seen in the image below, the orthocenter formed by 3 intersecting lines or altitudes lies outside the triangle.
Property 3: The orthocenter lies on the vertex of the right angle of the right triangle. As seen in the image below, the point of intersection lies at point C.
Property 4: An orthocenter divides an altitude into different parts. The product of the lengths of all these parts is equivalent for all three perpendiculars.
How to Construct an Orthocenter?
To construct the orthocenter for a triangle geometrically, we have to do the following:
- Find the perpendicular from any two vertices to the opposite sides.
- To draw the perpendicular or the altitude, use vertex C as the center and radius equal to the side BC. Draw arcs on the opposite sides AB and AC.
- Draw intersecting arcs from B and D, at F. Join CF.
- Similarly, draw intersecting arcs from points C and E, at G. Join BG.
- CF and BG are altitudes or perpendiculars for the sides AB and AC respectively.
- The intersection point of any two altitudes of a triangle gives the orthocenter.
- Thus, find the point of intersection of the two altitudes.
- At that point, H is referred to as the orthocenter of the triangle.
Orthocentre of a Triangle Formula
The orthocenter formula helps in locating the coordinates of the orthocenter of a triangle. Let us consider a triangle PQR, as shown in the figure below.
PA, QB, RC are the perpendicular lines drawn from the three vertices P[(x)1, (y)1], Q[(x)2, (y)2], and R[(x)3, (y)3] respectively of the △PQR.
H ( x, y) is the intersection point of the three altitudes of the triangle.
Step 1: Calculate the slope of the sides of the triangle using the formula:
m(slope) = \( \frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}} \)
Let slope of PR be given by mPR.
Hence,
mPR = \( \frac{ y_3 \ - \ y_1 }{ x_3 \ - \ x_1 }\)
Similarly,
mQR = \( \frac{ y_3 \ - \ y_2 }{ x_3 \ - \ x_2 }\)
Step 2 The slope of the altitudes of the △PQR will be perpendicular to the slope of the sides of the triangle.
We know,
\( \begin{align*} \text {Perpendicular slope of line} \ &= \ \frac{-1}{ \text {slope of the line}} \\ &= \frac{-1}{ \text m} \end{align*}\)
The slope of the respective altitudes:
Slope of PA, mPA = \( \frac{-1}{ \text {mQR}}\)
Slope of QB, mQB = \( \frac{ -1}{ \text {mPR}}\)
We will use the slope-point form equation as a straight line to calculate the equations of the lines coinciding with PA and QB.
The generalized equation thus formed by using arbitrary points (x) and (y) is:
\( \begin{align*} \text {mPA} &= \frac{( y \ - \ y_1 )}{( x \ - \ x_1 )} \\ \text {mQB} &= \frac{( y \ - \ y_2)}{( x \ - \ x_2 )} \end{align*} \)
Thus, solving the two equations for any given values the orthocenter of a triangle can be calculated.
Related Topics
Listed below are a few topics related to orthocenter, take a look.
Orthocenter Examples
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Example 1: Can you help Sam name the vertices, sides, altitudes, and orthocenter for the following figure?
Solution:
In the above figure for \( \triangle{ABC} \),
Vertices: A, B, and C
Sides: AB, BC, AC
Altitudes: AD, BE, CF
Orthocenter: H
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Example 2: Point H is the orthocenter of \(\triangle{ABC}\). If \(\angle{B} \ = \ 58^\circ , \ \angle{C}=60^\circ , \ \text {and} \angle{A}=62^\circ\). What is the measure of \(\angle{CHD}\)?
Solution:
In the given figure,
\( \begin{align*} \angle{ \text{CDH}} &= \angle{ \text{BFC}} = 90^\circ \\ \angle{ \text{DCH}} &= \angle{ \text {BCF}} \end{align*} \)
Hence, by AAA property,
\( \triangle{ \text {DCH}}\) and \( \triangle{ \text {FCB}} \) are similar triangles.
Thus,
\( \angle{ \text {CHD}} = \angle{ \text {CBF}} = 58^\circ \)
Therefore, \( \angle{ \text {CHD}} = 58^\circ\).
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Example 3:
Can you help Emma find the slopes of the altitudes of \( \triangle{ \text {ABC}}\) when its vertices are A (-5, 3), B (1, 7), C (7, -5)?
Solution:
Given, the vertices of the triangle,
\( \begin{align*} \text A &= (-5, 3) \\ \text B &= (1, 7) \\ \text C &= (7, -5) \end{align*} \)
Slope of AB,
\( \begin{align*} \text {mAB} &= \frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}} \\ &= \frac{7 - 3}{1 + 5} \\ &= \frac{4}{6} \\ &= \frac{2}{3} \end{align*} \)
Slope of CF,
\( \begin{align*} \text {mCF} &= \text {Perpendicular slope of AB} \\ &= \frac{-1}{ \text {mAB}} \\ &= \frac{-3}{2} \end{align*} \)
Slope of BC,
\( \begin{align*} \text {mBC} &= \frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}} \\ &= \frac{-5 - 7}{7 - 1} \\ &= \frac{-12}{6} \\ &= -2 \end{align*} \)
Slope of AD,
\( \begin{align*} \text {mAD} &= \text {Perpendicular slope of BC} \\ &= \frac{-1}{ \text {mBC}} \\ &= \frac{1}{2} \end{align*} \)
Slope of AC,
\( \begin{align*} \text {mAC} &= \frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}} \\ &= \frac{-5 - 3}{7 + 5} \\ &= \frac{-8}{12} \\ &= \frac{-2}{3} \end{align*} \)
Slope of BE,
\( \begin{align*} \text {mBE} &= \text {Perpendicular slope of AC} \\ &= \frac{-1}{ \text {mAC}} \\ &= \frac{3}{2} \end{align*} \)
Therefore, the slopes of the altitude is \( \frac{-3}{2} , \frac{1}{2} , \frac{3}{2}\).
FAQs on Orthocenter
What is an Orthocenter of a Triangle?
An orthocenter of a triangle is the point of intersection of altitudes that are drawn perpendicular from the vertex to the opposite sides of a triangle. A triangle usually has 3 altitudes and the intersection of all 3 altitudes is called the orthocenter. The placement of an orthocentre depends on the type of triangle it is. For example, an obtuse triangle has an orthocenter outside the triangle. An orthocenter is usually denoted by H.
What are the Properties of an Orthocenter?
The properties of an orthocenter of a triangle is different depending on the type of triangle, hence the properties are:
- The orthocenter lies inside the triangle for an acute angle triangle.
- The orthocenter lies outside the triangle for an obtuse angle triangle.
- The orthocenter lies on the vertex of the right angle of the right triangle.
- An orthocenter divides an altitude into different parts. The product of the lengths of all these parts is equivalent for all three perpendiculars.
What is the Orthocenter Formula?
The formula to calculate the orthocenter of a triangle of the given points is:
\( \begin{align*} \text {mPA} &= \frac{( y \ - \ y_1 )}{( x \ - \ x_1 )} \\ \text {mQB} &= \frac{( y \ - \ y_2)}{( x \ - \ x_2 )} \end{align*} \)
Why is it Called an Orthocenter?
The term "ortho" means "right" and the center means the midpoint. Thus, clubbing the two words together here means center for the altitudes (right angles) of the triangle. Hence, it is called an orthocenter.
Why is Orthocenter Important?
The orthocenter of a triangle is the point of intersection of all the three altitudes drawn from the vertices of a triangle to the opposite sides. It is an important central point of a triangle and thus helps in studying different properties of a triangle with respect to sides, vertices, other important points like circumcenter, centroid, etc.
Are Orthocenter and Circumcenter the Same?
No, the orthocenter and circumcenter of a triangle are different. The orthocenter of a triangle is the point of intersection of all the three altitudes drawn from the vertices of a triangle to the opposite sides. The circumcenter of a triangle is the point of intersection of the perpendicular bisector of the three sides.
What is the Difference Between Orthocenter and Incenter?
An incenter is a point where three angle bisectors from three vertices of the triangle meet. That point is also considered as the origin of the circle that is inscribed inside that circle. Whereas an orthocenter is a point where three altitudes of the triangle intersect.
What is the Difference Between Centroid, Orthocenter, Circumcenter, and Incenter?
A circumcenter is a point that is equidistant from all the vertices of the triangle and it is denoted as O. An incenter is the point that is equidistant from the sides of the triangle and it is denoted as I. An orthocenter is a point where all the altitudes of the triangle intersect and it is denoted as H. A centroid is the point of inspection of the medians of the triangles and it is denoted by G.