# 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)