Perpendicular Bisector to a Line Segment

overview

In this page, the following are explained.

• constructing a bisector to a line segment

• constructing a perpendicular through a point on a line

• constructing a perpendicular to a point not on a line

• constructing a bisector to a given angle

Perpendicular Bisector to a Line Segment

A rhombus is "A four sided figure with all sides equal".

A rhombus has the following properties.

• All sides are equal (and parallel)

• *the diagonals perpendicularly bisect*

• opposite angles are equal

• adjacent angles are supplementary

"Diagonals of rhombus perpendicularly bisect" This geometrical properties help to construct a perpendicular bisector to a line.

Given line segment ¯AB¯¯¯¯¯¯AB. The perpendicular bisector ¯pq¯¯¯¯pq is to be constructed.

• Take a compass

• fix a random distance between tips

• construct arcs from position AA above and below ¯AB¯¯¯¯¯¯AB

• construct arcs from position BB above and below ¯AB¯¯¯¯¯¯AB

• the intersecting points are PP and Q.

Note that the distance between tips of the compass is not modified and so, the sides ¯AP, ¯PB, ¯BQ, and ¯QA form a rhombus. From the property of a rhombus, the diagonals ¯AB and ¯PQ perpendicularly bisect each other.

*Note for curious students: It can be a kite, with ¯AB as minor diagonal. Property of a kite, the major diagonal ¯PQ bisects minor diagonal ¯AB. *

**Bisecting a Line Segment** : Use a compass to mark a rhombus with the given line segment as one of the diagonals. The perpendicular bisector is the other diagonal.

Perpendicular through a point on a line

Set squares or set triangles have sides or edges that are perpendicular.

Given line ¯AB and a point P on the line. Place the vertex of perpendicular edges of a set-square with one edge on the given line ¯AB. Construct the line along the other perpendicular edge of the set-square.

**Perpendicular through a point on a line** : Use the perpendicular edges of a set square, one edge on the given line and the perpendicular-vertex on the given point. Construct a line along the other perpendicular edge.

Some alternate methods can be used to construct a perpendicular through a point on a line.

• method to construct perpendicular bisector using a compass

• method to measure 90∘ angle using a protractor

**Perpendicular through a point on a line** : Use a compass to mark points L and M on the line that are in equal distance from the given point P. Construct the perpendicular bisector of ¯LM.

**Perpendicular through a point on a line** : Place the protractor with origin on the given point and the base-line on the given line. Mark a point on 90∘ angle and draw a line through the given point and the marked point.

Perpendicular to a point not on a line

Set squares can help to construct a perpendicular to a line through a point outside the line.

**Perpendicular to a Point** : Use the perpendicular edges of a set square, one edge on the given line and the other edge on the given point. Construct a line along the other perpendicular edge.

**Perpendicular to a Point** : Use a compass to mark points L and M on the line that are in equal distance from the given point P. Construct the perpendicular bisector of ¯LM.

Bisecting an angle

A kite is a quadrilateral with two pair of equal and adjacent sides.

A kite has the following properties

• two pair of equal sides

• major diagonal perpendicularly bisects the minor diagonal. *the diagonal that divides the kite into two congruent triangles is called major. The other diagonal is the minor diagonal.*

• *major diagonal bisects the angles at the vertices*

• two equal opposite angles and two unequal opposite angles -- all sum up to 360∘

The geometrical property "major diagonal of a kite bisect the angles at both vertices" can be used to bisect a given angle.

Given angle ∠BAC, a kite AQRP is formed.

• With an arbitrary measure on compass, mark points P and Q from the point A.

• With another arbitrary measure on compass, construct arcs from points P and Q. These two arcs cut at point R.

The quadrilateral AQRP is a kite and the line ¯AR is the major bisector of angle ∠BAC.

**Bisecting an Angle** : Using the two rays of the angle, create a kite such that the major diagonal bisects the angle.

summary

**Bisecting a Line Segment** : Use a compass to mark a rhombus with the given line segment as one of the diagonals. The perpendicular bisector is the other diagonal.

**Perpendicular through a point on a line** : Use the perpendicular edges of a set square, one edge on the given line and the perpendicular-vertex on the given point. Construct a line along the other perpendicular edge.

**Perpendicular through a point on a line** : Use a compass to mark two points on the line that are in equal distance from the given point. Construct the perpendicular bisector between the two marked points.

**Perpendicular through a point on a line** : Place the protractor with origin on the given point and the base-line on the given line. Mark a point on 90∘ angle and draw a line through the given point and the marked point.

**Perpendicular to a Point** : Use the perpendicular edges of a set square, one edge on the given line and the other edge on the given point. Construct a line along the other perpendicular edge.

**Perpendicular to a Point** : Use a compass to mark points two points on the line that are in equal distance from the given point. Construct the perpendicular bisector between the two marked points.

**Bisecting an Angle** : Using the two rays of the angle, create a kite such that the major diagonal bisects the angle.

Outline

The outline of material to learn "Construction / Practical Geometry at 6-8th Grade level" is as follows.
Note: * click here for detailed outline of "constructions / practical geometry".*

• Four Fundamenatl elements

→ __Geometrical Instruments__

→ __Practical Geometry Fundamentals__

• Basic Shapes

→ __Copying Line and Circle__

• Basic Consustruction

→ __Construction of Perpendicular Bisector__

→ __Construction of Standard Angles__

→ __Construction of Triangles__

• Quadrilateral Forms

→ __Understanding Quadrilaterals__

→ __Construction of Quadrilaterals__

→ __Construction of Parallelograms__

→ __Construction of Rhombus__

→ __Construction of Trapezium__

→ __Construction of Kite__

→ __Construction of Rectangle__

→ __Construction of Square__