In geometry, the concept of distance plays a crucial role in determining the length between two points on a coordinate plane. This fundamental concept is essential in various fields, including mathematics, physics, and engineering. The distance between two points is defined as the length of the line segment that connects the two given points.
Understanding the distance between two points is essential for various applications, such as calculating the shortest route between two locations, measuring the distance between objects, or determining the length of a line segment. In coordinate geometry, the distance between two points can be calculated using the distance formula, which is an application of the Pythagorean theorem.
In this article, we will explore the concept of distance between two points in depth. We will discuss the definition of distance, the distance formula, its derivation, and how to find the distance between two points of coordinates. Additionally, we will explore the distance between two points in a complex plane and three-dimensional space. Throughout the article, we will provide examples to illustrate the concepts and formulas discussed.
What is Distance?
Distance is a numerical measurement that represents the length between two points or objects. In mathematics, distance is a fundamental concept used to measure the separation between two points in a given space. It is a scalar quantity, meaning it only has magnitude and no direction. The distance between two points is always positive and cannot be negative.
Distance can be measured in various units, such as meters, kilometers, miles, or any other unit of length. It is important to note that the unit of measurement depends on the context and the specific application.
What is Distance Between Two Points?
The distance between two points is the length of the line segment that connects the two given points. In a coordinate plane, the distance between two points can be calculated by finding the length of the line segment joining the given coordinates. The distance between two points is always positive and represents the shortest path between the two points.
The distance between two points is denoted by the symbol “d” and is calculated using the distance formula. The distance formula calculates the distance between two points using their coordinates in a coordinate plane.
Distance Formula
The distance formula is a mathematical formula used to find the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In the context of the distance formula, the two sides of the right-angled triangle represent the horizontal and vertical distances between the two points, and the hypotenuse represents the distance between the two points.
The distance formula for two points (x1, y1) and (x2, y2) in a coordinate plane is given by the following equation:
d = √((x2 – x1)² + (y2 – y1)²)
where d represents the distance between the two points.
Derivation of Formula for Distance Between Two Points of Coordinates
To derive the formula to calculate the distance between two points in a two-dimensional plane, let’s assume that there are two points with the coordinates given as A(x1, y1) and B(x2, y2). Next, we will assume that the line segment joining A and B is AB = d. Now, let’s plot the given points on the coordinate plane and join them by a line.
Next, we will construct a right-angled triangle with AB as the hypotenuse. Applying the Pythagorean theorem for the triangle ABC, we can derive the distance formula.
AB² = AC² + BC² d² = (x2 – x1)² + (y2 – y1)²
Taking the square root on both sides, we obtain the distance formula:
d = √((x2 – x1)² + (y2 – y1)²)
Thus, the distance formula to find the distance between two points is proved.
How to Find Distance Between Two Points of Coordinates?
To find the distance between two points using their coordinates, we can follow these steps:
- Note down the coordinates of the two given points in the coordinate plane as A(x1, y1) and B(x2, y2).
- Apply the distance formula: d = √((x2 – x1)² + (y2 – y1)²).
- Simplify the equation and calculate the distance.
- Express the answer in the appropriate units based on the given coordinates.
Let’s consider an example to understand how to find the distance between two points of coordinates.
Example: Find the distance between the points A(1, 2) and B(4, 5).
Solution: Given the coordinates of the points A and B as (1, 2) and (4, 5) respectively, we can use the distance formula to calculate the distance between them.
d = √((x2 – x1)² + (y2 – y1)²) = √((4 – 1)² + (5 – 2)²) = √(3² + 3²) = √(9 + 9) = √18 ≈ 4.24 units
Therefore, the distance between the points A(1, 2) and B(4, 5) is approximately 4.24 units.
Distance Between Two Points in Complex Plane
In addition to the two-dimensional coordinate plane, the concept of distance between two points can also be extended to the complex plane. The complex plane is a two-dimensional plane where each point is represented by a complex number of the form a + bi, where a and b are real numbers and i is the imaginary unit (√(-1)).
The distance between two points in the complex plane is calculated using a formula similar to the distance between two points formula in the Cartesian plane. Consider two complex numbers z1 = a + bi and z2 = c + di. Recall the fact that every complex number on a complex plane corresponds to a point on the coordinate plane. Then, the distance between the two complex numbers z1 and z2 is given by:
|z1 – z2| = √((a – c)² + (b – d)²)
Here, |z1 – z2| represents the absolute value of the complex number z1 – z2.
Let’s consider an example to find the distance between two complex numbers.
Example: Find the distance between the complex numbers z1 = 1 + 3i and z2 = 2 – 4i.
Solution: The points that denote the given complex numbers are (1, 3) and (2, -4). So, the distance between them is:
|z1 – z2| = √((1 – 2)² + (3 + 4)² = √(1 + 49) = √50 ≈ 5√2 units
Distance Between Two Points Formula in 3D Space
In three-dimensional space, the distance between two points is calculated using the three-dimensional distance formula. The formula is an extension of the distance formula in the two-dimensional plane, considering the additional coordinate in the z-axis.
Let’s consider two points A(x1, y1, z1) and B(x2, y2, z2) in three-dimensional space. The distance between these two points is given by the following formula:
d = √((x2 – x1)² + (y2 – y1)² + (z2 – z1)²)
Here, d represents the distance between the two points.
The three-dimensional distance formula can be derived by extending the derivation of the two-dimensional distance formula, considering the additional coordinate in the z-axis.
Distance between Two Points: Using Pythagoras’ Theorem
Another way to find the distance between two points is by using Pythagoras’ theorem. Pythagoras’ theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In the context of finding the distance between two points, we can consider the horizontal and vertical distances between the two points as the two sides of a right-angled triangle. The distance between the two points is then equal to the length of the hypotenuse of this triangle.
Let’s consider an example to illustrate the use of Pythagoras’ theorem to find the distance between two points.
Example: A boy started from point A and walked west for 12 miles. He then turned to the north and walked for 5 miles more. Find the shortest distance between the initial position and the final position.
Solution: To find the shortest distance between the initial position and the final position, we can consider a right-angled triangle formed by the two distances traveled by the boy.
Let’s denote the initial position as A and the final position as C. The distance between points A and B is 12 miles, and the distance between points B and C is 5 miles.
To find the shortest distance, we can use Pythagoras’ theorem:
AC² = AB² + BC² AC² = 12² + 5² AC² = 144 + 25 AC² = 169 AC = √169 AC = 13 miles
Therefore, the shortest distance between the initial position and the final position is 13 miles.
What Are the Coordinates of a Point?
In coordinate geometry, the position of a point is defined by its coordinates along the x-axis and y-axis. The coordinates of a point represent its location in the coordinate plane. The x-coordinate, also known as the abscissa, represents the distance of the point from the y-axis. The y-coordinate, also known as the ordinate, represents the distance of the point from the x-axis.
In the Cartesian coordinate system, the coordinates of a point are written as an ordered pair (x, y), where x represents the distance from the y-axis, and y represents the distance from the x-axis. For example, the point A(3, 4) represents a point that is 3 units away from the y-axis and 4 units away from the x-axis.
Coordinates of a point on the x-axis are of the form (x, 0), where x is the distance of the point from the origin. Coordinates of a point on the y-axis are of the form (0, y), where y is the distance of the point from the origin.
Distance of a Point from the Origin
The distance of a point from the origin is the length of the line segment joining the point and the origin. In a two-dimensional plane, the distance of a point P(x, y) from the origin O(0, 0) is given by the formula:
OP = √(x² + y²)
Here, OP represents the distance between the point P(x, y) and the origin O(0, 0).
To find the distance, we can substitute the coordinates of the point into the formula and calculate the square root of the sum of the squares of the coordinates.
Solved Examples on How to Use the Distance Between Two Points
Let’s solve some examples to understand how to use the distance between two points formula.
Example 1: Find the distance between the two points A(1, 2) and B(4, 5).
Solution: Given the coordinates of the points A and B as (1, 2) and (4, 5) respectively, we can use the distance formula to calculate the distance between them.
d = √((x2 – x1)² + (y2 – y1)²) = √((4 – 1)² + (5 – 2)²) = √(3² + 3²) = √(9 + 9) = √18 ≈ 4.24 units
Therefore, the distance between the points A(1, 2) and B(4, 5) is approximately 4.24 units.
Example 2: Show that the points A(1, 7), B(4, 2), C(-1, -1), and D(-4, 4) are the vertices of a square.
Solution: Let A(1, 7), B(4, 2), C(-1, -1), and D(-4, 4) be the coordinates of the four points in an XY plane.
To prove that these points form a square, we need to show that the lengths of the sides AB, BC, CD, and AD are equal. Additionally, we need to show that the lengths of the diagonals AC and BD are equal.
We can use the distance formula to calculate the lengths of the sides and diagonals:
AB = √((4 – 1)² + (2 – 7)²) = √(3² + (-5)²) = √(9 + 25) = √34 BC = √((-1 – 4)² + (-1 – 2)²) = √((-5)² + (-3)²) = √(25 + 9) = √34 CD = √((-4 + 1)² + (4 – (-1))²) = √((-3)² + 5²) = √(9 + 25) = √34 AD = √((4 + 1)² + (2 – (-1))²) = √(5² + 3²) = √(25 + 9) = √34
AC = √((1 – (-1))² + (7 – (-1))²) = √(2² + 8²) = √(4 + 64) = √68 BD = √((4 – (-4))² + (2 – 4)²) = √(8² + (-2)²) = √(64 + 4) = √68
Since AB = BC = CD = AD and AC = BD, we can conclude that the points A(1, 7), B(4, 2), C(-1, -1), and D(-4, 4) form a square.
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In conclusion, the distance between two points is a fundamental concept in geometry that plays a crucial role in various applications. Understanding the distance between two points and how to calculate it using the distance formula is essential for solving problems in coordinate geometry. By applying the formulas and concepts discussed in this article, you can confidently calculate the distance between any two points in a coordinate plane.