Vertices of a Triangle

What are Vertices of a Triangle

In geometry, a vertex (plural vertices) is a point where two straight lines intersect. A triangle is formed by the intersection of three line segments. Each side of a triangle has two endpoints, with the endpoints of all three sides meeting at three different points in a plane, forming a triangle. The three different intersecting points or corners are called the vertices of a triangle.

How Many Vertices Does a Triangle Have

A triangle has three vertices or corners.

△ABC is formed by three line segments

• Side AB and AC intersect at point A, So, A is a vertex
• Side AB and BC intersect at point B, So, B is a vertex
• Side AC and BC intersect at point C, So, C is a vertex

Thus, in △ABC, ‘A’, ‘B’, and ‘C’ are the 3 vertices

Formula

How to Find the Vertices of a Triangle If the Midpoints of Its Sides are Given

If (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of the midpoints of the 3 sides of a triangle, then we can find the vertices using the following formula:

Let us solve an example to understand the concept better.

Solved Example

(4, 1), (2, 5) and (3, 1) are the midpoints of the sides of a triangle. Find the vertices of the given triangle.

Solution:

Let A, B and C are three vertices of the given triangle.
As we know,
Vertex A = (x₁ + x₃ – x₂, y₁ + y₃ – y₂)
Vertex B = (x₁ + x₂ – x₃, y₁ + y₂ – y₃)
Vertex C = (x₂ + x₃ – x₁, y₂ + y₃ – y₁)
Given,
x₁ = 4, y₁ = 1
x₂ = 2, y₂ = 5
x₃ = 3, y3 = 1
Putting the values we get,
Vertex A = (x₁ + x₃ – x₂, y₁ + y₃ – y₂)
= (4 + 3 – 2, 1 +1 – 5)
 = (5, -3)
 Vertex B = (x₁ + x₂ – x₃, y₁ + y₂ – y₃)
 = (4 + 2 – 3, 1 + 5 -1)
 = (3, 5)
Vertex C = (x₂ + x₃ – x₁, y₂ + y₃ – y₁)
 = (2 + 3 – 4, 5 + 1 – 1)
 = (1, 5)
Thus, the vertices of the given triangle are:
A = (5, -3), B = (3, 5), and C = (1, 5)