What is a Vertex in Geometry?
Last Updated :
30 Dec, 2023
When two or more lines are met at a point to form an angle, that point can be called a vertex. So vertex can be defined as a point when two or more lines meet to form an angle. A vertex is denoted by capital letters like A, B, E etc. In geometry there are many shapes like cube, square, triangle etc. For these figures, there are more than one vertex. So the plural form of a vertex is called vertices. Let’s look at a few figures
There is an Euler’s formula to calculate how many vertices are present for a three dimensional (3D) figure. The formula is given by-
Euler’s Formula-> F + V – E = 2
The above formula can be modified to get vertices count as
V = E + 2 – F
Where
V represents number of vertices
F represents number of faces
E represents number of edges
Let’s look at the few questions on finding the number of vertices for the given figures.
Question 1: Find the number of vertices present for a figure (cube) with 6 faces and 12 edges.
Solution:
Given
Number of faces (F) = 6
Number of edges (E) = 12
From Euler’s Method,
Number of vertices (V) = E + 2 – F
= 12 + 2 – 6
= 8
So number of vertices for given figure is 8.
Question 2: Find the number of vertices present for a 3D cylinder that is having 2 faces (Top and Bottom are covered) and 0 edges.
Solution:
Given
Number of faces (F) = 2
Number of edges (E) = 0
From Euler’s Method,
Number of vertices (V) = Â E + 2 – F
= 0 + 2 – 2
= 0
So number of vertices for given figure is 0.
Vertex of Parabola
In Parabola, the vertex is a point where it actually turns. This is also called a minimum point/maximum point. When the parabola opens down the vertex is called as maximum point else minimum point.
There are two ways of finding the vertex in a parabola based on the given form of the equation.
If the given equation of a parabola is of form ax2+bx+c then vertex of the parabola is given by-
V = (-b/2a, -D/4a)
WhereÂ
D = b2 – 4ac
If the given equation of a parabola is of form y = a(x – h)2 + k, then the vertex of the parabola is given by-
V = (h , k)
Let’s look into a few examples in finding the vertex of the parabola.
Question 1: Find the vertex of parabola if the equation of parabola is 3x2 + x – 2.
Solution:
GivenÂ
Equation of parabola 3x2+x-2
It is of form ax2+bx+c where a=3, b=1, c=-2
So vertex of parabola is V=(-b/2a,-D/4a)
Discriminant can be calculate by formula D=b2-4ac
D=12-4×3×(-2)
=1-(-24)
=1+24
D=25
Vertex (V)=(-b/2a,-D/4a)
=(-1/2(3),-25/4(3))
V=(-1/6,-25/12)
Hence the vertex of parabola 3x2+x-2 is at (-1/6,-25/12)
Question 2: What is the vertex of parabola if the equation of parabola is x2-4x+3.
Solution:
Given
Equation of parabola is x2-4x+3
It is of form ax2+bx+c where a=1, b=-4, c=3
So vertex of parabola is V=(-b/2a,-D/4a)
Discriminant can be calculate by formula D=b2-4ac
D=(-4)2-4×1×3
=16-12
D=4
Vertex (V)=(-b/2a,-D/4a)
=(-(-4)/2(1),-4/4(1))
=(4/2,-4/4)
V=(2,-1)
Hence the vertex of parabola x2-4x+3 is at (2,-1)
Question 3: Find the vertex of parabola if the equation of parabola is y = 3(x-4)2+2.
Solution:
Given
Equation of parabola is y=3(x-4)2+2
It is of form y=a(x-h)2+k where a=3, h=4, k=2
So vertex of parabola is V=(h,k)
Vertex (V)=(h , k)
=(4,2)
Hence the vertex of parabola 3(x-4)2+2 is at (4,2)
Question 4: what is the vertex of parabola if the equation of parabola is y = 2x2-8x+9.
Solution:
Given
Equation of parabola is y=2x2-8x+9
This can be rewritten into y=2(x-2)2+1
It is of form y=a(x-h)2+k where a=2, h=2, k=1
So vertex of parabola is V=(h,k)
Vertex (V)=(h , k)
=(2,1)
Hence the vertex of parabola 2x2-8x+9 is at (2,1)
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