Grade 8 → Coordinate Geometry ↓
Distance Between Points
In this topic, we will learn how to calculate the distance between two points in the plane using coordinate geometry. Understanding how to find this distance is an important concept in mathematics, often encountered in real-world scenarios such as navigation and mapping.
Understanding coordinates
To begin, let's review the coordinate system. In a two-dimensional plane, each point is identified by an ordered pair of numbers, usually written as (x, y). The first number x represents the horizontal position, and the second number y represents the vertical position.
The coordinate plane has two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which is denoted by (0, 0).
Distance formula
The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula, which is derived from the Pythagorean theorem.
Distance formula
D = √((x2 - x1)² + (y2 - y1)²)
Let's break down this formula:
(x2 - x1)is the difference of the horizontal positions (x-coordinates) of the two points.(y2 - y1)is the difference of the vertical positions (y-coordinates) of two points.- We square these differences, add them up, and then take the square root of the sum to find the distance.
Deriving the distance formula
To understand where this formula comes from, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Imagine a right triangle where:
- The horizontal side is
(x2 - x1)(difference in x-coordinates). - The vertical side is
(y2 - y1)(the difference in the y-coordinate).
The hypotenuse of this triangle represents the distance between the two points. Applying the Pythagorean theorem, we get:
c² = a² + b²,
Where:
c = Distance (d),
a = (x2 - x1),
b = (y2 - y1).
From this, we get the formula:
d² = (x2 – x1)² + (y2 – y1)²
=> D = √((x2 - x1)² + (y2 - y1)²)
Visualise the concept
Let's look at a visual representation of how to calculate the distance between two points using a formula. Suppose we have two points A and B with coordinates A(x1, y1) and B(x2, y2).
In this diagram, the black line between A and B represents the distance we want to calculate.
Example calculation
Let's calculate the distance between points A(1, 2) and B(4, 6) using the distance formula:
given:
Point A (x1, y1) = (1, 2)
Point B (x2, y2) = (4, 6)
Distance, d = √((x2 - x1)² + (y2 - y1)²)
= √((4 - 1)² + (6 - 2)²)
= √(3² + 4²)
= √(9 + 16)
= √25
= 5
Therefore, the distance between the two points is 5 units.
More examples
Let's look at a few more scenarios to strengthen our understanding:
Example 1
Find the distance between the points C(2, 3) and D(5, 7).
given:
The point C (x1, y1) = (2, 3)
Point D (x2, y2) = (5, 7)
Distance, d = √((x2 - x1)² + (y2 - y1)²)
= √((5 - 2)² + (7 - 3)²)
= √(3² + 4²)
= √(9 + 16)
= √25
= 5
The distance is 5 units.
Example 2
Find the distance between the points E(-3, -5) and F(1, 2).
given:
The point E (x1, y1) = (-3, -5)
The point F (x2, y2) = (1, 2)
Distance, d = √((x2 - x1)² + (y2 - y1)²)
= √((1 - (-3))² + (2 - (-5))²
= √((1 + 3)² + (2 + 5)²)
= √(4² + 7²)
= √(16 + 49)
= √65
≈ 8.06
The distance is approximately 8.06 units.
Special cases
Horizontal and vertical lines
When two points lie on a horizontal or vertical line, the distance formula becomes simple.
Horizontal line
If the points have the same y coordinate, the distance will simply be the difference in the x-coordinates.
For example, for the points G(3, 4) and H(7, 4):
Distance, d = √((x2 - x1)²)
= |x2 - x1|
= |7 - 3|
= 4
The distance is 4 units, since they lie on a horizontal line.
Vertical line
If x coordinates of the points are the same, the distance will simply be the difference of the y-coordinates.
For example, for the points I(6, 1) and J(6, 5):
Distance, d = √((y2 - y1)²)
= |y2 - y1|
= |5 - 1|
= 4
The distance is 4 units, since they lie on a vertical line.
Applications of distance formula
Distance formulas are widely used not only in mathematics but also in various fields such as physics, engineering, computer graphics, and navigation. Here are some examples:
Geography and mapping
In navigation and mapping, the distance formula helps determine the shortest path between two geographic points. This can be important for determining routes, planning logistics, or locating services.
Computer graphics
In computer graphics, calculating the distance between two points is fundamental to rendering and simulating environments. It allows for detecting collisions, creating realistic shadows, and maintaining scene proportions.
Physics
Physics often relies on calculating distance to understand motion and forces. The distance between two points over time is important in analyzing speed, velocity, and acceleration.
Conclusion
Calculating the distance between two points on the coordinate plane is a fundamental tool in mathematics. Whether you're working in purely mathematical contexts or applying it to real-world problems, knowing how to use the distance formula can make your work simpler and clearer.
Practicing this concept through examples and exercises deepens one's understanding and ability to apply it effectively in different scenarios. The more one engages in these calculations, the more intuitive and immediate they become.
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