Explanation
In two dimensions \(2D\), points lie on a flat surface called the coordinate plane, also known as the \(xy\)-plane.
Each point is described by two coordinates:
$$(x,y)$$
The \(x\)-coordinate gives the horizontal position, and the \(y\)-coordinate gives the vertical position.
If we know the coordinates of two points, we can use the distance formula in \(2D\) to find the straight-line distance between them.
The distance formula comes from the Pythagorean Theorem. The horizontal and vertical changes form the two sides of a right triangle, while the distance between the two points forms the hypotenuse.
Formula / Rule
If the two points are:
$$P_1=(x_1,y_1)$$
and
$$P_2=(x_2,y_2)$$
then the distance between them is:
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Where:
- \(d\) is the distance between the two points.
- \(x_1\) and \(y_1\) are the coordinates of the first point.
- \(x_2\) and \(y_2\) are the coordinates of the second point.
The distance is always non-negative.
Example
Find the distance between:
$$P_1=(1,2)$$
and
$$P_2=(4,6)$$
Step 1: Use the distance formula.
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Step 2: Substitute the coordinates.
$$d=\sqrt{(4-1)^2+(6-2)^2}$$
Step 3: Simplify.
\begin{align}
d &= \sqrt{3^2+4^2}\\
\\
&= \sqrt{9+16}\\
\\
&= \sqrt{25}\\
\\
&=5
\end{align}
So, the distance between the two points is \(5\) units.
Video Explanation
Practice
Continue Learning
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