Distance formulae
Understanding Distance Formulae
Distance formulae are mathematical equations used to determine the distance between two points in various geometrical spaces. These formulae are fundamental in the study of geometry, trigonometry, and other mathematical fields. In this content, we will explore the distance formulae in the context of a two-dimensional Cartesian plane, which is the most common application in high school mathematics and standardized exams.
Distance Formula in a Cartesian Plane
The distance between two points in a two-dimensional space can be calculated using the Pythagorean theorem. If we have two points, $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $d$ between these points is given by:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
This formula is derived from the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical distances between the points.
Example:
Calculate the distance between the points $A(3, 4)$ and $B(7, 1)$.
Using the distance formula:
$$ d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $$
The distance between points A and B is 5 units.
Distance from a Point to a Line
Another important concept is finding the distance from a point to a line in a Cartesian plane. The formula for the distance $D$ from a point $P(x_0, y_0)$ to a line given by the equation $Ax + By + C = 0$ is:
$$ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} $$
Example:
Find the distance from the point $P(2, -3)$ to the line $3x + 4y - 5 = 0$.
Using the formula:
$$ D = \frac{|3(2) + 4(-3) - 5|}{\sqrt{3^2 + 4^2}} = \frac{|6 - 12 - 5|}{\sqrt{9 + 16}} = \frac{|-11|}{\sqrt{25}} = \frac{11}{5} = 2.2 $$
The distance from point P to the line is 2.2 units.
Comparing Distance Formulae
Let's compare the two distance formulae we've discussed:
| Aspect | Distance Between Two Points | Distance from Point to Line |
|---|---|---|
| Formula | $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ | $D = \frac{ |
| Application | Used to find the distance between two points in a plane | Used to find the shortest distance from a point to a line |
| Derived From | Pythagorean theorem | Perpendicular distance from a point to a line |
| Requires | Coordinates of two points | Coordinates of a point and the equation of a line |
| Result | Length of the line segment connecting the points | Length of the perpendicular from the point to the line |
Important Points to Remember
- The distance formula for two points is symmetric; swapping the points does not change the distance.
- The distance from a point to a line is always the shortest distance; it is the length of the perpendicular from the point to the line.
- The distance calculated using these formulae is always a non-negative value.
- When using the distance from a point to a line formula, ensure that the line's equation is in the standard form $Ax + By + C = 0$.
By understanding these distance formulae and their applications, students can solve a wide range of geometric problems involving distances in a Cartesian plane. These concepts are not only crucial for academic exams but also form the foundation for more advanced studies in mathematics and related fields.