Explanation
In two dimensions \(2D\), each point on the coordinate plane is located using two coordinates:
$$(x,y)$$
If we know the coordinates of two points, we can find the straight-line distance between them using the distance formula.
The distance between two points is the length of the line segment connecting them.
We can find this distance in two ways:
- algebraically, using the distance formula
- graphically, by forming a right triangle and using the Pythagorean Theorem
Both methods give the same result.
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.
◉ \(x_2-x_1\) gives the horizontal change.
◉ \(y_2-y_1\) gives the vertical change.
Remember:
- Distance is never negative.
- The order of the two points does not change the distance.
- Squaring the differences removes negative signs.
- The formula comes from the Pythagorean Theorem.
Example
Find the distance between:
$$P_1=(3,7)$$
and
$$P_2=(-5,1)$$
Algebraic Method
Step 1: Write the distance formula.
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Step 2: Substitute the coordinates.
$$d=\sqrt{(-5-3)^2+(1-7)^2}$$
Step 3: Simplify the differences.
$$d=\sqrt{(-8)^2+(-6)^2}$$
Step 4: Square the values.
\begin{align}
d &= \sqrt{64+36}\\
\\
&= \sqrt{100}\\
\\
&= 10\\
\end{align}
✅ So, the distance between the two points is \(10\) units.
Graphical Idea
The horizontal distance between the two points is:
$$8$$
and the vertical distance is:
$$6$$
These form the two sides of a right triangle.
The distance between the points is the hypotenuse.
Using the Pythagorean formula:
\begin{align}
d &= 8^2+6^2\\
\\
&= 64+36\\
\\
&= 100\\
\\
&= 10
\end{align}
So, both methods give the same distance:
$$d=10\text{ units}$$
Video Explanation
Practice
Continue Learning
- Plotting Points on the Coordinate Plane (\(2D\))
- Which Quadrant or Axis? Points on the Coordinate Plane \(2D\)
- Distance Formula in \(2D\)
- Distance between Two Points in \(2D\) (Current lesson)
- Distance Formula: Radius and Area of a Circle