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 (x_{1}, y_{1}), (x_{2}, y_{2}), and (x_{3}, y_{3}) 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 = (x_{1} + x_{3} – x_{2}, y_{1} + y_{3} – y_{2}) Vertex B = (x_{1} + x_{2} – x_{3}, y_{1} + y_{2} – y_{3}) Vertex C = (x_{2} + x_{3} – x_{1}, y_{2} + y_{3} – y_{1}) Given, x_{1} = 4, y_{1} = 1 x_{2} = 2, y_{2} = 5 x_{3} = 3, y3 = 1 Putting the values we get, Vertex A = (x_{1} + x_{3} – x_{2}, y_{1} + y_{3} – y_{2}) = (4 + 3 – 2, 1 +1 – 5) = (5, -3) Vertex B = (x_{1} + x_{2} – x_{3}, y_{1} + y_{2} – y_{3}) = (4 + 2 – 3, 1 + 5 -1) = (3, 5) Vertex C = (x_{2} + x_{3} – x_{1}, y_{2} + y_{3} – y_{1}) = (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)