How To Find The Vertex Of A Circle
Vertex is the point where two or more lines or edges meet to form an angle. In this lesson, you will learn about vertex examples, the vertex definition, and the vertex angle.
Let's have a look at the two rays: Ray 1 and Ray 2, they originate from the vertex O.
As we go forward, you can check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.
Learn all you need to know about vertex in this short lesson!
Lesson Plan
What Is a Vertex?
A vertex in math is a point where two lines or rays meet.
An angle is formed at the vertex.
A vertex is denoted by uppercase letters like A, O, P, etc.
The plural of vertex is "vertices." In solid geometry, i.e., three-dimensional geometry, shapes such as cubes, cuboids form several vertices. Let us look at them in detail in the next section.
Look at the triangle DEF.
it has three vertices: D, E, and F that form the angles of the triangle.
Look at this pentagon below.
Can you count the number of vertices in the given pentagon?
What Is a Vertex In Solid Geometry?
Not just plane shapes, even solid shapes have vertices.
In solid shapes, a vertex is formed where edges meet.
Examples
Look at this cube,
A, B, C, D, E, F, G, H are the vertices of the cube.
A tetrahedron has 4 vertices. One is marked. Can you identify the other 3 vertices of the tetrahedron?
We can apply Euler's formula to find the vertices in a solid shape.
Euler's formula is:
Where,
- F is the number of faces
- V stands for the vertices
- E is the number of edges
What Is the Vertex of a Parabola?
When we graph a quadratic equation, we get a parabola.
The vertex definition of a parabola is the point where exactly it turns.
It is also called the minimum point.
When the parabola , opens down the vertex is called the maximum point.
The parabola vertex lies at the axis of symmetry.
In the standard form, we write the quadratic equation as \(ax^2 + bx + c\)
In the standard form, the vertex (V) of the parabola is given by:
\[V \equiv \left( { - \frac{b}{{2a}}, -\frac{D}{{4a}}} \right)\]
where D is the discriminant. \(\begin{align}x = {-b \pm \sqrt{b^2-4ac} \over 2a}\end{align} \)
The vertex equation of a parabola is of the form \(y = a(x-h)^2 + k\)
The vertex of the parabola is at the coordinate (h, k)
Vertex of the parabola (h,k)
Example 1
Graph of \(f\left( x \right) = 1 - 2x -3{x^2}\) is as shown below . Find the vertex of the parabola.
Solution
We have:
\[\begin{array}{l}a = - 3,\,\,\,b = - 2,\,\,\,c = 1\\ \Rightarrow \,\,\,D = {b^2} - 4ac =16\end{array}\]
The coordinates of the vertex are.
\[V \equiv \left( { - \frac{b}{{2a}}, -\frac{D}{{4a}}} \right) = \left( { - \frac{1}{3},\frac{4}{3}} \right)\]
Note that the parabola will open downward (since \(a\) is negative), but the vertex has a positive y-coordinate.
\(\therefore\) Vertex of the parabola is at \(-\dfrac{1}{3} , \dfrac{4}{3}\)
Example 2
Find the vertex of the parabola \(y =2(x+3)^2−8\)
Solution
We know that
The vertex equation of a parabola is of the form \(y = a(x – h)^2 + k\) where (h,k) is the vertex.
In the given equation, h = 3 and k = - 8
Therefore the vertex of the parabola is (3,-8)
\(\therefore\) Vertex V of the parabola is at (3,-8)
Important Notes
- The number of vertices in a solid shape can be found using Euler's formula F + V − E = 2
- The vertex of the parabola \(ax^2 +bx +c\) is \(V \equiv \left( { - \frac{b}{{2a}}, -\frac{D}{{4a}}} \right)\)
- The vertex form of the parabola equation is \(y = a(x – h)^2 + k\) where (h, k) is the vertex.
Solved Examples
How many vertices does a rectangle have?
Solution
Vertices are the corners of the rectangle.
A rectangle has 4 corners, and hence it has 4 vertices.
\(\therefore\) A rectangle has 4 vertices.
How many vertices does a circle have?
Solution
A circle does not have any corner.
Hence, there are no vertices in a circle.
\(\therefore\) A circle does not have any vertices.
Tarun said there are as many vertices in a Rubik's cube as in a cuboid.
Is Tarun correct?
Solution
Let us count the number of corners in the Rubik's cube and the cuboid.
Both of them have 8 vertices.
\(\therefore\) Tarun is correct. A cuboid and a Rubik's cube have 8 vertices.
Help Manav identify the number of vertices in each of the given shapes.
Solution
We can count the number of corners in each of the shapes to tell the number of vertices.
| Shape | Number of Vertices |
|---|---|
| Cylinder | 0 |
| Square pyramid | 5 |
| Sphere | 0 |
Challenging Questions
- Does a cone have a vertex?
- How many vertices will a polygon of n sides have?
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about vertex with the examples and practice questions. Now you will be able to easily solve problems on vertex definition, vertex example, vertex angle, and vertex in a parabola.
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Frequently Asked Questions(FAQs)
1. What is the meaning of vertex?
Vertex is the point of intersection of edges or line segments.
2. Why is the vertex important?
Vertex is important as it defines the high point or low point, as the high point in an isosceles triangle or the minimum point in a parabola.
3. What is the vertex of a graph?
Vertex is the node of the graph. It is also the point that defines the node or the maximum or minimum point.
4. What is the vertex of an angle example?
The vertex of an angle is the point where two rays begin or meet.
Look at the example: Ray1 and Ray 2 originate from vertex O.
How To Find The Vertex Of A Circle
Source: https://www.cuemath.com/geometry/vertex-definition/
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