Perpendicular Bisector Calculator

Formula

To calculate the Perpendicular Bisector:

$Y - Y_1 = m \times (X - X_1)$

Where:

$m$ is the slope, calculated as $\frac{Y_2 - Y_1}{X_2 - X_1}$
$Y_2$ and $Y_1$ are the two y coordinates
$X_2$ and $X_1$ are the two x coordinates

What is a Perpendicular Bisector?

A perpendicular bisector is a line or line segment that cuts another line segment into two equal parts at a right angle. It is created by finding the midpoint of the line segment and then constructing a line perpendicular to it.

One significant application of the perpendicular bisector is in triangle geometry. When three perpendicular bisectors are drawn in a triangle, they intersect at a single point known as the circumcenter. The circumcenter is the circle’s center that passes through all three vertices of the triangle. This property is fundamental in a variety of geometric proofs and constructions involving triangles.

Additionally, the perpendicular bisector is used in determining the location of the centroid of a triangle. The centroid is the point of intersection of the three medians, the line segments connecting each vertex of the triangle to the midpoint of the opposite side.

Example Calculation 1

Let's assume the following values:

X1 = 2
Y1 = 3
X2 = 8
Y2 = 7

Using the formula:

Midpoint:

$\left( \frac{2 + 8}{2}, \frac{3 + 7}{2} \right) = (5, 5)$

Slope:

$m = \frac{7 - 3}{8 - 2} = \frac{4}{6} = \frac{2}{3}$

Perpendicular Slope:

$m_{\perp} = -\frac{1}{m} = -\frac{3}{2}$

Equation of Perpendicular Bisector:

$Y - 5 = -\frac{3}{2}(X - 5)$