Definition and Properties of a Point with Examples
According to modern mathematics, a point usually refers to an element of some set known as space. Going more in-depth, in Euclidean geometry, a point is a primitive notation upon which the geometry is built. It means that a point cannot be defined in terms of the previously defined objects. Point is defined by some of the properties known as axioms and that must be satisfied. In specific, point geometry will not have length, volume, area or any other dimensional attribute.
The dots shown in the above diagram is known as a point, but a point has really no size.
Cartesian Coordinates
The exact location of the point can be shown using the cartesian coordinates. For clear understanding here we will see an example the point (12,5)
In point (12,5), X-axis is 12 units long and Y-axis is 5 units away from the centre. The left and right direction or horizontal direction is known as X-axis. And the up and down direction or vertical direction is known as the X-axis.
Points in the Euclidean Geometry
Points is considered as the framework of Euclidean geometry and it is one of the most fundamental objects. Euclid defined a point as “ that which has no part”, in two-dimensional space, points are represented as ordered pairs (x,y) of numbers. Conventionally, this first line is represented in the horizontal line and the second line is represented on the vertical line. The idea can be easily assumed that three-dimensional Euclidean space is represented in the ordered triplet (x,y,z) in this edition of the third number represents the depth and denoted by z.
Did you know?
Depending on the coordinates of the points, we can find the quadrant in which the point lies.
Like if the point is negative on the x-axis and positive for the y axis, thus the point must lie in the 2nd quadrant. And if the point is positive on the x-axis and negative on the y axis, thus the point lies in the 4th quadrant. If the point is negative in both the axis, thus the point lies in the third quadrant. And if the point is positive in both the axis, thus the point lies in the first quadrant.
Now, solve an example for proper understanding.
Q. Find in which quadrant the point (-3,4) lies.
Depending on the above-given statements, the point is negative on the x-axis and the point is positive on the y-axis so the point lies in the second quadrant.
Let us See Some of the Solved Problems Related to Point in Math.
1. Find the Distance Between the Two Points A(4,3), B(8,6).
Answer. The distance between the two points A(4,3) and B(8,6).
Assuming the first point as x\[_{1}\], y\[_{1}\] and the second point as x\[_{2}\], y\[_{2}\].
To find the distance between the two points we have the formula AB = \[\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\]
Substituting all the values in the formula given above,
We get, = \[\sqrt{(8 - 4)^{2} + (6 - 3)^{2}}\]
= \[\sqrt{(4)^{2} + (3)^{2}}\]
= \[\sqrt{16+9}\]
= \[\sqrt{25}\]
= 5 units
So, the distance between the two points A and B is 5 units.
2. Find the Coordinates of the Point on the X-axis, Which is at a Distance of 5 Units from the Point (6,-3).
Answer. Let us assume that the coordinates of the points on the X-axis is (x,0).
The distance formula is, D = \[\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\]
Now, substitute the given values in the formula.
We get, 5 = \[\sqrt{(x - 6)^{2} + (0 - (- 3))^{2}}\]
implies, 25 = x\[^{2}\] - 12x + 36 + 9
Solving further we get, x=2 or x=10.
So the required points are (2,0) and (10,0).
FAQs on What Is a Point in Geometry
1. What is a point in mathematics?
A point in mathematics is an exact location in space that has no size, no length, no width, and no thickness. It only indicates position. In geometry, a point is usually represented by a dot and named using a capital letter such as A, B, or P. A point is one of the fundamental building blocks of geometry, along with lines and planes.
2. How is a point represented in geometry?
A point is represented by a dot and labeled with a capital letter. For example:
- A dot marked with the letter A is called Point A.
- In coordinate geometry, a point is written as an ordered pair like (x, y).
3. What are the coordinates of a point?
The coordinates of a point describe its exact position on the coordinate plane using numbers. In a 2D Cartesian plane, a point is written as (x, y), where:
- x is the horizontal position (x-coordinate).
- y is the vertical position (y-coordinate).
4. What is the difference between a point and a line?
A point shows a single location, while a line is a straight path extending infinitely in both directions. The key differences are:
- A point has no dimension.
- A line has one dimension (length).
- A line is made up of infinitely many points.
5. What is the distance between two points formula?
The distance between two points in a coordinate plane is given by the formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]. For example, for points (1, 2) and (4, 6):
- d = √[(4 − 1)² + (6 − 2)²]
- d = √[3² + 4²]
- d = √[9 + 16] = √25 = 5
6. What is the midpoint of two points?
The midpoint of two points is the point exactly halfway between them. The midpoint formula is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). For example, for points (2, 4) and (6, 8):
- M = ((2 + 6)/2, (4 + 8)/2)
- M = (8/2, 12/2)
- M = (4, 6)
7. Can a point exist without dimensions?
Yes, a point has zero dimensions in geometry. It has:
- No length
- No width
- No height
8. What is the origin in coordinate geometry?
The origin is the point where the x-axis and y-axis intersect, and its coordinates are (0, 0). It serves as the reference point for measuring positions on the Cartesian plane. All other points are located based on their distance and direction from the origin.
9. How do you plot a point on a graph?
To plot a point (x, y) on a graph, move along the x-axis first and then along the y-axis. Follow these steps:
- Start at the origin (0,0).
- Move right if x is positive or left if x is negative.
- Move up if y is positive or down if y is negative.
- Mark the final position with a dot and label it.
10. What are collinear points?
Collinear points are points that lie on the same straight line. If three or more points share a single straight line, they are collinear. For example:
- Points (1,1), (2,2), and (3,3) are collinear.
- They lie on the line with equation y = x.
