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Axis in Coordinate Geometry

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In the Geometry, the points on the plane are defined by using a pair of Numbers and they are called coordinates.
Suppose we have a table, and the table has 6 columns which are named as P, Q, R, S, T, U and six rows are present in the table which is 1, 2, 3, and so on. And we have value ‘w’ in the box R3; it means ‘R’ column and 3 rows, where ‘R’ and 3 are said to be the coordinates of a table.
Similarly points can be plotted on axes with help of coordinates. Now we will talk about Axis of Symmetry;
Two Axis in coordinate geometry are present in a plane and they are known as x – axis and y – axis.
The axis along the horizontal direction is known as x – axis and the axis along the vertical direction is said to be y – axis or in other words we can say that along the x –axis a perpendicular is drawn that perpendicular is known as y – axis.
In case of x – axis the values present to the right side are positive values and the values present on the left side are negative values.
In case of y – axis the values present above the origin are positive values and the values present below the origin are negative values. The location of a Point in a plane of geometry is given by the pair of numbers. In pair the first number tells that where it lies on the x – axis and the second number tells that where it lies on the y – axis. It doesn’t represent two locations for the single variable.
Both axis ‘x’ and ‘y’ define the unique Position in the plane of geometry. This is all about the axis of coordinate geometry.

Axis of Symmetry

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A line which passes through a shape in such a way that each side is mirror image of each other is known as Axis of Symmetry. We can say that a line which is divides the figure in two symmetrical parts and each part is mirror image of each other part is known as Axis of Symmetry.
Now we will see types of symmetry.
There are three types of symmetry which are shown below:
x – Axis symmetry.
y – axis symmetry,
Origin symmetry.

To find the axis of symmetry we have to follow some steps which are shown below:
Step 1: To find the axis of symmetry we take equation of figure.
Step 2: Then identify the coordinates of given expression.
Step 3: Now axis of symmetry can be found by using the formula shown below:
Axis of symmetry = -b/2a;
Step 4: Put the values in the given formula we get axis of symmetry.
Now we will understand these steps with the help of an example:
Suppose we have an expression y = x2 + 14x + 34 and we have to find the axis of symmetry.
To find the axis of symmetry we have to follow steps shown below:
Step 1: First we take an expression y = x2 + 14x + 34;
Then identify the coordinates of an expression:
Let a = 1, b = 14 and c = 34;
We know that the formula for finding the axis of symmetry is:
Axis of symmetry = -b/2a;
Put the value of ‘a’ and ‘b’ in the formula:
Axis of symmetry = -14/2 (1);
Axis of symmetry = -14/2;
So axis of symmetry is -7.
This is how we can find the axis of symmetry.

Axis of Rotation

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In order to study about axis of rotation, we will first look at the term rotation and Rotational Symmetry. When any two dimensional figure is moved by certain angle, at the axis which does not change its place. Suppose we take the example of fan, we say that Centre of fan remains fixed and three wings of fan rotates around the fixed Point. So this fixed point around which any figure rotates is called as axis of rotation.

On rotating the figures around its axis, we sometimes come to the situations where we find the figure looks exactly same as original figure. Then this point is called as point of rotational symmetry of the particular figure. Suppose we take a Circle, in circle, we say that centre of circle is the axis of rotation of circle. So we say that circle if rotated by any of the angle will be at the rotational symmetry.

This simply means that point about which the particular figure is rotated, then we say that it is the axis of the figure.
Now we will look at axis of rotation of Square. We observe that meeting point of two diagonals of square is the axis of rotation of square. It signifies that if we rotate the square about this point of rotation, we can find 4 positions where figure looks exactly same. Thus we conclude that square has the rotational symmetry of order 4. In order to find the angle of rotation of figure, we will divide the measure 360 degrees by the order of rotation.
Thus we conclude that angle of rotation of square will be equals to 360/ 4 = 90 degrees.

Axis of a Solid

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Geometry is defined as branch of mathematics which deals with different shapes and figures which we come across in our day to day life. These figures are Solid figures like Cube, cuboids, cone, square, rectangle, sphere, cylinder and many more.

Each solid has its symmetry that is an axis along which if figure is rotated then it appears to be same. Axis of solid can be defined as an imaginary line passing through any solid about which or we can say around which a figure is symmetrical. Thus, axis of a solid is defined for every figure and it’s not mandatory that each figure has only one Axis of Symmetry it can be more than one and can be number of axes of symmetry.

Let us consider a cylinder, axis of symmetry in a cylinder can be considered as a line passing through center of cylinder crossing both ends of base, this line will be axis of symmetry as if cylinder is rotated along this axis then it appears to be same. That is along this axis, axis of rotation appears same. Similarly, a cube has also axis of symmetry but it has number of axes, that is a cube may possess different types of axes of symmetry that are, six 2 fold axes which appears as passing through midpoints of two opposite edges, second type is four 3 fold axes which pass through two opposite vertices and last type is of three 4 fold axes which is passing through two opposite vertices.

Thus, in total a cube can have 13 axes of symmetry. Similarly, a tetrahedron has seven axes of rotation or we can say axes of symmetry. An icosahedrons possess thirty one axes of symmetry and a Prism contains n + 1 axes of symmetry.