Vertex is a special point that helps us identify different types of triangles. It is the third element, along with side and angle, which makes a triangle different than any other polygon. Vertex often appears in geometry problems as well as in real-life scenarios like climbing, hill walking, and rock climbing. Vertex can be found using the standard form of a triangle, which is also known as a right triangle. Let’s see how to find the vertex from the standard form of a triangle so that you can understand its usage in various activities.

**How To Find Vertex From Standard Form?**

- Write the standard form of a vertex v in the graph G. This can be done by finding the lowest index number of v’s neighbors, which are non-terminal nodes and are not v. The standard form of the graph G with vertex v is given by the following expression.
- From the expression (1), we can get the standard form of a vertex v from its neighbors’ indices. The following is a table of necessary steps for finding a vertex v from its neighbors’ indices.
- To find the standard form of edge e, and to find all vertices in G, we must know how to write an expression that represents the edges and vertices in G. This can be done by writing a list of expressions that represent each edge and vertex in G, as shown in below table:
- So far in this section, we have found the standard forms of nodes and edges in graph G by using simple methods such as finding the lowest index number of non-terminal nodes and non-adjacent edges, respectively. We have not explained how to find the standard forms of vertices or cycles. In this section, we will introduce some different methods to find them, which are easier than those used so far. We will start with simple methods first and then explain more complicated methods.
- Find the Standard Form of a Vertex from Its Neighbors’ Indices If a vertex v has no neighbors, then its standard form is simply v. By doing this, we can find the standard forms of all vertices in G.
- Find the Standard Form of an Edge from Its Neighbors’ Indices If an edge e has no neighbors, then its standard form is simply e. By doing this, we can find the standard forms of all edges in G.
- Find the Standard Form of a Vertex with Two Neighbors If a vertex v has two neighbors u and w, then their lowest index numbers are u and w, respectively (where 0 < u < w). To find their standard forms using simple methods such as finding lowest index number of non-adjacent edges and non-adjacent vertices, respectively, we must know how to write expressions that represent each neighbor pair in G as shown below:
- Here we have seen how to find the standard forms of vertices with two neighbors. The following is a table that shows how to find the standard forms of vertices with two neighbors by using simple methods.
- Here we have seen how to find the standard forms of vertices with two neighbors. The following is a table that shows how to find the standard forms of vertices with two neighbors by using simple methods.
- Find the Standard Form of a Vertex from Its Neighbors’ Indices and Its Nodes If a vertex v has no neighbors, then its standard form is simply v, but we can also use its lowest index number as its standard form. In this case, we can use its lowest index number as both its standard form and neighbor index number for finding all vertices in G.

**What Is A Vertex?**

In geometry, a vertex is the first element of a triangle. It is the point where two sides of a triangle meet. The third element of a triangle is an angle. The point where two angles meet is called a vertex. Every triangle has at least one vertex. Each vertex has a letter associated with it depending on which side of the triangle touches it. The letters for the vertices of a triangle are A, B, and C. These letters are used to describe the shape of a triangle as well as for naming the sides of a triangle. The side opposite a vertex is called an adjacent side. The side adjacent to a vertex is called the opposite side. The longest side of a triangle is called the hypotenuse. The shortest side is called the adjacent side, while the side in between the shortest and longest is called the opposite side.

**Finding Vertex In The Standard Form Of A Triangle**

- To find the vertex of a triangle, we have to find the side opposite to it. From the figure above, we can see that the side opposite C is B. This is because B touches C at point P.
- The second step is to find which side touches another side. In this example, B touches A at point Q and D touches A at point R. So we need to find which side D touches A on. The answer is C because D touches C on QPR and C touches A on PQ.
- The third step is to find the point where another side touches a vertex. In this example, D touches B on point R. So we need to find the point where C touches B on. The answer is P because C touches B on QPR and D touches B on point R.
- To complete the triangle, we need to find the opposite side of the triangle. In this example, A is opposite to D and E is opposite to A. So from the figure above, we can see that the answer is E.
- The final step is to find the hypotenuse of the triangle. From the figure above, we can see that B is the hypotenuse because it touches all three vertices at their respective points.
- Now we have found all the sides of a triangle and we have found its vertex.
- To find the length of a side, we need to find how long an angle is in the triangle and then use Pythagoras’ theorem to find the length of that side:
- The length of the side opposite a vertex is called the adjacent side, while the side opposite a vertex is called the opposite side. The sides adjacent to a vertex are called opposite sides.
- To find the length of an angle, we just need to find how long an arc is in the triangle and then use Pythagoras’ theorem to find its length:
- The length of the angle opposite a vertex is called the adjacent angle, while the angle adjacent to a vertex is called its opposite angle. The angles adjacent to a vertex are called opposite angles.

**Conclusion**

The article explains in detail what a vertex is, how to find vertex using the standard form of a triangle, and why it is important to find the location of the vertex of a triangle. If you want to solve geometry problems, then knowing how to find vertex from the standard form of a triangle is very essential.