Length of a Line

Distance Formula

The length of a straight line between two points on a coordinate plane can be found using the distance formula. If you have two points with coordinates $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between them is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

This formula comes from applying Pythagoras' theorem to the horizontal and vertical distances between the points.

For example, if you have points $A(3, 4)$ and $B(7, 1)$, the length of the line $AB$ can be calculated by finding the differences in coordinates, squaring them, adding, and then taking the square root.

For instance, if you want to find the length of the line between points $A(3, 4)$ and $B(7, 1)$, first calculate the differences in the x- and y-coordinates:

• $x_2 - x_1 = 7 - 3 = 4$
• $y_2 - y_1 = 1 - 4 = -3$

Then square these differences and add:

$$4^2 + (-3)^2 = 16 + 9 = 25$$

Finally, take the square root:

$$d = \sqrt{25} = 5$$

So, the length of the line $AB$ is $5\,\mathrm{units}$. Note that "units" depends on the scale of the coordinate plane.

Application of Length in Coordinate Geometry

The distance formula is very useful in coordinate geometry problems, especially when working with shapes drawn on the coordinate plane. Here are some common applications:

Finding the Perimeter of Shapes

To find the perimeter of a polygon on a coordinate plane, calculate the length of each side using the distance formula and then add all the lengths together.

For example, to find the perimeter of a triangle with vertices at $P(1, 2)$, $Q(4, 6)$, and $R(7, 2)$, find the lengths of sides $PQ$, $QR$, and $RP$ and sum them.

Checking Equal Lengths in Geometry

You can use the distance formula to check if two line segments are equal in length, which helps in identifying special types of triangles (like isosceles) or verifying if a quadrilateral is a square or rectangle.

For example, if two sides of a triangle have the same length, the triangle is isosceles.