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In the field of geometry, a plane is defined as a level surface that goes on forever.
A two-dimensional surface is another name for this type of surface.
A plane is characterised by its absence of thickness and curvature as well as its infinite breadth and length.
It is difficult to conceptualise a plane being in the real world.
Nonetheless, the flat surfaces of a cube or cuboid, as well as the flat surface of a sheet of paper, are all examples of real-world instances of geometric planes.
An example of a plane is presented here in which the location of any particular point on the plane may be located by employing a certain ordered pair of numbers, often known as coordinates.
The coordinates provide an accurate representation of the points’ locations on the plane.
The Characteristics of Planes
In mathematics, a plane possesses the following characteristics:
If there are two separate planes, then those planes must either be parallel to one another or they must intersect one another along a line.
Either a line is perpendicular to a plane, meaning that it crosses the plane at exactly one point, or the line is contained within the plane itself.
In the event that there are two separate lines, each of which is perpendicular to the same plane, then those lines must be parallel to one another.
It follows that any two distinct planes that are perpendicular to the same line must be parallel to one another.
Equation of a plane in normal form
If you have the normal vector of a plane and a point that passes across the plane, you may determine the equation of the plane as follows: a (x– x1) + b (y– y1) + c (z– z1) = 0.
A point that can be described as being perpendicular to a particular course.
Two geometric objects are considered perpendicular in elementary geometry if they intersect at a right angle, defined as 90 degrees or π/2 radians.
To create the perpendicular to the line AB that passes through the point P using a compass and a straightedge, carry out the steps that are outlined below (see picture on the left):
The first step is to build a circle with its centre at P.
This will result in the creation of points A’ and B’ on line AB which is the same distance from P.
The second step is to create circles of the same radius that are centred at A’ and B’. Allow Q and P to represent the points where these two circles meet one another.
In the third step, you will join Q and P in order to create the needed perpendicular PQ.
If a line meets multiple lines in a plane and all of those lines are perpendicular to each other, then we may say that the line in question is perpendicular to the plane.
This concept is dependent on how perpendicularity between lines is understood.
Three points that are not collinear are given
If you draw a circle with O as the centre and OP as the radius, then the circle will also go through Q and R.
Due to the fact that the perpendicular bisectors of PQ and QR only cross at O, point O is the only point that is equidistant from points P, Q, and R.
As a result, the letter O will serve as the circle’s centre when it is drawn.
The radii of the circle will be OP, OQ, and OR respectively.
Given that the points do not lie in a straight line, it follows that it is possible to design a unique circle that passes through all three of those points.
Up until this point, you have learnt how to design a circle that passes through three points that are not colinear.
Examples
Question:
What is the equation of the circle that has
the points A (-2, 0), B (0, 0), and C (-2, 0) as its points of intersection?
Solution:
Think about the circle’s equation in its most general form:
x2 + y2 + 2gx + 2fy + c = 0…. (i)
Applying the substitution A(2, 0) in (i),
(2)2 + (0)2 + 2g(2) + 2f(0) + c = 0 4 + 4g + c = 0….(ii)
Applying the substitution B(-2, 0) in (i),
(-2)2 + (0)2 + 2g(-2) + 2f(0) + c = 0 4 – 4g + c = 0….(iii)
Putting in the expression C(0, 2) in (i),
(0)2 + (2)2 + 2g(0) + 2f(2) + c = 0 4 + 4f + c = 0….(iv)
Adding (ii) and (iii),
4 + 4g + c + 4 – 4g + c = 0 2c + 8 = 0 2c = -8 c = -4
Using the equation c = -4 in (ii),
4 + 4g − 4 = 0
4g = 0,
g = 0
Using the equation c = -4 in (iv),
4 + 4f – 4 = 0
4f = 0 f = 0
Changing the values of g, f, and c in this expression now (i),
x2 + y2 + 2(0)x + 2(0)y + (-4) = 0 x2 + y2 – 4 = 0 Or x2 + y2 = 4
This is the equation for the circle that goes through the three points that have been supplied to us: A, B, and C.
Conclusion
A surface that is flat and stretches eternally in two dimensions but does not have any thickness is called a plane.
Because there is nothing in the real world that may serve as a true example of a geometric plane, it can be a little challenging to get a mental picture of what a plane looks like.
However, we can depict a portion of a geometric plane by using the surface of a wall, the floor, or even a sheet of paper if we choose to do so.
You only need to keep in mind that, in contrast to the parts of planes that exist in the real world, geometric planes do not have any edges.
