Orthocenter vs. Centroid — What's the Difference?
By Maham Liaqat & Fiza Rafique — Updated on May 15, 2024
The orthocenter is the point where the altitudes of a triangle intersect, while the centroid is the point where the medians of a triangle intersect and also acts as the triangle's center of mass.
Difference Between Orthocenter and Centroid
Table of Contents
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Key Differences
The orthocenter of a triangle is the point where the three altitudes meet. An altitude is a perpendicular line drawn from a vertex to the opposite side (or its extension). The orthocenter can be inside, outside, or on the triangle, depending on the type of triangle. The centroid of a triangle, on the other hand, is the point where the three medians intersect. A median is a line segment drawn from a vertex to the midpoint of the opposite side.
Both the orthocenter and centroid are significant in triangle geometry but serve different purposes. The orthocenter relates to the altitudes and their perpendicularity, while the centroid is concerned with the medians and the balance of the triangle.
Comparison Chart
Definition
Intersection of the altitudes of a triangle
Intersection of the medians of a triangle
Construction
Formed by drawing perpendiculars from vertices to opposite sides
Formed by connecting vertices to midpoints of opposite sides
Position
Can be inside, outside, or on the triangle
Always inside the triangle
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Triangle Type Influence
Location depends on whether the triangle is acute, obtuse, or right
Always inside regardless of triangle type
Function
Relates to perpendicularity and altitude lengths
Represents the center of mass or balance point
Compare with Definitions
Orthocenter
Intersection point of the altitudes of a triangle.
The orthocenter of an acute triangle lies inside the triangle.
Centroid
Point where each median divides into a 2:1 ratio.
The centroid divides each median into two segments, one of which is twice as long as the other.
Orthocenter
Key point in triangle geometry related to altitudes.
The orthocenter helps in understanding triangle heights and their intersections.
Centroid
Always lies inside the triangle.
Regardless of the triangle's type, the centroid is always inside.
Orthocenter
Location varies with triangle type.
In a right triangle, the orthocenter is at the right-angle vertex.
Centroid
Intersection point of the medians of a triangle.
The centroid of a triangle is the balance point of the triangle.
Orthocenter
Point where perpendiculars from each vertex meet.
To find the orthocenter, draw altitudes from each vertex to the opposite side.
Centroid
Represents the center of mass or balance point.
If you were to balance a triangle on a pin, it would balance perfectly at the centroid.
Orthocenter
Can lie inside, outside, or on the triangle.
The orthocenter of an obtuse triangle lies outside the triangle.
Centroid
Important in geometry for dividing a triangle into smaller triangles of equal area.
The centroid creates six smaller triangles within the original triangle, all with equal area.
Orthocenter
The point of intersection of the three altitudes of a triangle.
Centroid
In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions.While in geometry the word barycenter is a synonym for centroid, in astrophysics and astronomy, the barycenter is the center of mass of two or more bodies that orbit each other.
Orthocenter
(geometry) The intersection of the three lines that can be drawn flowing from the three corners of a triangle to a point along the opposite side where each line intersects that side at a 90 degree angle; in an acute triangle, it is inside the triangle; in an obtuse triangle, it is outside the triangle.
Centroid
See center of mass.
Orthocenter
That point in which the three perpendiculars let fall from the angles of a triangle upon the opposite sides, or the sides produced, mutually intersect.
Centroid
The point in a system of masses whose coordinates on each dimension are a weighted mean of the coordinates of points on that dimension, the weights being determined by the density function of the system.
Centroid
The point at which gravitational force (or other universally and uniformly acting force) may be supposed to act on a given rigid, uniformly dense body; the centre of gravity or centre of mass.
Centroid
The point of intersection of the three medians of a given triangle; the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of the three vertices.
Centroid
(of a finite set of points) the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of a given finite set of points.
Centroid
An analogue of the centre of gravity of a nonuniform body in which local density is replaced by a specified function (which can take negative values) and the place of the body's shape is taken by the function's domain.
The centroid of an arbitrary function is given by , where the integrals are calculated over the domain of .
Centroid
The arithmetic mean (alternatively, median) position of a cluster of points in a coordinate system based on some application-dependent measure of distance.
Centroid
Given a tree of n nodes, either (1) a unique node whose removal would split the tree into subtrees of fewer than n/2 nodes, or (2) either of a pair of adjacent nodes such that removal of the edge connecting them would split the tree into two subtrees of exactly n/2 nodes.
Centroid
The center of mass, inertia, or gravity of a body or system of bodies.
Centroid
The center of mass of an object of uniform density
Common Curiosities
Can the centroid be outside the triangle?
No, the centroid always lies inside the triangle.
How is the centroid constructed?
The centroid is constructed by drawing line segments (medians) from each vertex to the midpoint of the opposite side.
What is the significance of the centroid in triangle geometry?
The centroid is significant as it represents the triangle's balance point and is used in various geometric calculations.
What is the orthocenter of a triangle?
The orthocenter is the point where the three altitudes of a triangle intersect.
Can the orthocenter be outside the triangle?
Yes, in obtuse triangles, the orthocenter lies outside the triangle.
What is the centroid of a triangle?
The centroid is the point where the three medians of a triangle intersect and acts as the triangle's center of mass.
What is the significance of the orthocenter in triangle geometry?
The orthocenter is significant for understanding the properties related to the altitudes and their perpendicular intersections.
Does the orthocenter have a specific formula?
The orthocenter's coordinates can be found using the slopes of the altitudes, but there is no simple averaging formula like the centroid.
How is the orthocenter constructed?
The orthocenter is constructed by drawing perpendicular lines (altitudes) from each vertex to the opposite side.
Where is the orthocenter in an acute triangle?
In an acute triangle, the orthocenter lies inside the triangle.
Where is the orthocenter in a right triangle?
In a right triangle, the orthocenter is located at the vertex of the right angle.
What is the ratio in which the centroid divides the medians?
The centroid divides each median into a 2:1 ratio, with the longer segment being closer to the vertex.
Which point is known as the center of mass of the triangle?
The centroid is known as the center of mass of the triangle.
Do the orthocenter and centroid coincide in any triangle?
The orthocenter and centroid coincide in an equilateral triangle, where all significant points of concurrency (orthocenter, centroid, circumcenter, and incenter) are the same.
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Written by
Maham Liaqat
Co-written by
Fiza RafiqueFiza Rafique is a skilled content writer at AskDifference.com, where she meticulously refines and enhances written pieces. Drawing from her vast editorial expertise, Fiza ensures clarity, accuracy, and precision in every article. Passionate about language, she continually seeks to elevate the quality of content for readers worldwide.
