Calculadora de bisectriz de segmento

The perpendicular bisector of a line segment is the line that crosses the segment at its midpoint AT A RIGHT ANGLE (90°). It is one of the most important constructions in geometry — it is the unique line consisting of all points equidistant from the segment's two endpoints. The Segment Bisector Calculator takes two endpoint coordinates and returns the perpendicular bisector's equation. This tutorial explains what makes the perpendicular bisector special, the equation derivation, and the broader role it plays in triangle geometry (circumcenter, perpendicular bisector theorem).

Two defining properties

A line is the perpendicular bisector of segment AB if BOTH of the following hold:

1. It passes through the midpoint of AB.
2. It is perpendicular (at 90°) to AB.

Either property alone is not enough. A line through the midpoint that isn't perpendicular is just "a bisector" (not perpendicular). A perpendicular line that doesn't pass through the midpoint is just "a perpendicular line" (not a bisector). The perpendicular bisector is the unique line satisfying both.

The equidistance property

The perpendicular bisector has a remarkable property — its locus definition:

A point lies on the perpendicular bisector of segment AB if and only if it is equidistant from A and B.

This is the Perpendicular Bisector Theorem. It means:

• Every point on the perpendicular bisector is equally distant from the two endpoints.
• Conversely, any point equidistant from A and B lies on the perpendicular bisector.

Geometrically: the perpendicular bisector is the SET OF ALL POINTS at equal distance from A and B. This locus characterization is why the perpendicular bisector appears in so many distance-based constructions.

Worked example — find the perpendicular bisector

Find the perpendicular bisector of the segment from A = (2, 1) to B = (8, 5).

Step 1: Find the midpoint.

M = ((2 + 8) / 2, (1 + 5) / 2) = (5, 3).

Step 2: Find the slope of AB.

m_AB = (5 − 1) / (8 − 2) = 4 / 6 = 2/3.

Step 3: Take the negative reciprocal for the perpendicular slope.

m_perp = −1 / (2/3) = −3/2.

Step 4: Write the equation in point-slope form.

y − 3 = (−3/2)(x − 5)

Or in slope-intercept form: y = (−3/2)x + 15/2 + 3 = y = (−3/2)x + 10.5.

Verification — check equidistance

Pick a point on the perpendicular bisector — say (5, 3) (the midpoint). Distance to A = √((5−2)² + (3−1)²) = √13. Distance to B = √((5−8)² + (3−5)²) = √13. Equal. ✓

Try another point. From the bisector equation, at x = 1: y = −1.5 + 10.5 = 9. Distance from (1, 9) to A = √(1 + 64) = √65. To B = √(49 + 16) = √65. Equal. ✓

The circumcenter

One of the four classical "triangle centers": the circumcenter is the point where the three perpendicular bisectors of a triangle's three sides meet. It is the center of the circumscribed circle — the unique circle passing through all three vertices.

Why all three meet at one point: each perpendicular bisector is the locus of points equidistant from two of the triangle's vertices. Where the bisector of side AB meets the bisector of side BC, the point is equidistant from A, B, AND C — so it lies on the bisector of side CA too. All three concurrence.

The circumcenter's distance to each vertex equals the circumradius R. For an obtuse triangle, the circumcenter lies OUTSIDE the triangle.

Compass-and-straightedge construction

The perpendicular bisector is one of the foundational compass-and-straightedge constructions:

1. Open your compass to any width greater than half the segment length.
2. Place the compass at one endpoint, draw an arc on both sides of the segment.
3. Without changing the compass width, place it at the other endpoint, draw an arc on both sides. The arcs intersect at TWO points (one above the segment, one below).
4. Use a straightedge to connect those two intersection points. This is the perpendicular bisector.

Why it works: by the compass setting, both intersection points are equidistant from the two endpoints (each was hit by both arcs at the same radius). By the perpendicular bisector theorem, those equidistant points lie on the perpendicular bisector, so the line through them IS the bisector.

The perpendicular slope formula

Two lines with slopes m₁ and m₂ are perpendicular if and only if:

m₁ × m₂ = −1

(equivalently, m₂ = −1/m₁ — the "negative reciprocal").

Special cases:

• Horizontal segment (slope 0): perpendicular slope is undefined → bisector is vertical.
• Vertical segment (slope undefined): perpendicular slope is 0 → bisector is horizontal.
• Slope 1: perpendicular slope is −1.

Real-world applications

• Locating equidistant facilities. A new fire station should be equidistant from two existing stations — locate it on the perpendicular bisector of the line segment between them.
• Mediation / fairness problems. Splitting a property line equidistant from two boundaries uses the perpendicular bisector concept.
• Computer graphics. Voronoi diagrams partition a plane based on distance to "seed" points; the boundaries between Voronoi cells are perpendicular bisectors of the seeds.
• GPS triangulation. Locating a position from distances to known points uses perpendicular bisector intersections.

Common mistakes

• Using the original slope instead of the negative reciprocal. The perpendicular bisector's slope is the NEGATIVE RECIPROCAL of the segment's slope, not the same.
• Forgetting it passes through the midpoint. A line perpendicular to AB but not through the midpoint isn't a "perpendicular bisector" — it's just a perpendicular line. Both conditions matter.
• Confusing perpendicular bisector with angle bisector. Different things: the perpendicular bisector is for line segments; the angle bisector divides an angle into two equal parts.
• Treating a vertical segment's "negative reciprocal" as undefined. A vertical segment has undefined slope; its perpendicular has slope 0 (horizontal). Use the special-case rule, not the formula.