Welcome to your ultimate and comprehensive study guide for Line Bisectors and Angle Bisectors. Geometry provides rigorous logical frameworks for proving spatial relationships. This chapter explores foundational geometric theorems and their converses concerning right bisectors of line segments, angle bisectors, and the concurrency of these bisectors in triangles.
1. Right Bisector of a Line Segment
A line is called a right bisector of a line segment if it is perpendicular to the line segment and passes through its midpoint.
- Theorem 1: Any point on the right bisector of a line segment is equidistant from its end points.
- Theorem 2 (Converse): Any point equidistant from the end points of a line segment is on the right bisector of it.
- Theorem 3: The right bisectors of the sides of a triangle are concurrent, intersecting at a single common point known as the circumcentre.
2. Bisector of an Angle
A ray BP is called the bisector of angle ABC if P is a point in the interior of the angle and the angle ABP is equal in measure to angle PBC.
- Theorem 4: Any point on the bisector of an angle is equidistant from its arms.
- Theorem 5 (Converse): Any point inside an angle, equidistant from its arms, is on the bisector of it.
- Theorem 6: The bisectors of the angles of a triangle are concurrent, intersecting at a single common point known as the incentre.
3. Step-by-Step Example: Equidistance Property
Example: Understanding the right bisector equidistance property.
Step-by-Step Solution:
- Step 1: Identify given properties: Let line LM intersect line segment AB at point C perpendicularly, such that AC equals BC, and P is any point on LM.
- Step 2: Form triangles: Join P to endpoints A and B to form triangles ACP and BCP.
- Step 3: Apply congruence: Since AC = BC, angle ACP = angle BCP (both 90 degrees), and PC is common, triangle ACP is congruent to triangle BCP by the SAS postulate.
- Step 4: Conclude distance: Therefore, side PA equals side PB, proving that any point on the right bisector is equidistant from the endpoints.
Essential Conceptual Review Questions
Q1: What is the point of concurrency of the right bisectors of the sides of a triangle called?
Answer: It is called the circumcentre of the triangle, which is equidistant from all three vertices.
Q2: Where does the point of concurrency of the angle bisectors of a triangle lie?
Answer: It lies inside the triangle and is called the incentre, which is equidistant from all three sides of the triangle.