To calculate the Perpendicular Bisector:
\[ Y - Y_1 = m \times (X - X_1) \]
Where:
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.
Let's assume the following values:
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:
\[ Y - 5 = -\frac{3}{2}(X - 5) \]