Bisector

Bisectors are lines or planes in geometry that split an angle, line segment, or shape into two equal parts. They are crucial in mathematics fields like geometry, trigonometry, and calculus. Bisectors help solve problems involving angles, triangles, circles, and other geometric figures by dividing them equally.

Introduction

Bisectors are a basic concept in geometry. They are lines, rays, or planes that split an angle, line segment, or shape into two equal parts. The term "bisect" comes from the Latin word "bisector," meaning "to cut into two equal parts." Bisectors are important in geometry because they help solve problems involving angles, triangles, and circles. They also help find the center of mass of a shape, which is useful in physics and engineering.

Definition

In geometry, a bisector is a line, ray, or plane that divides an angle, line segment, or shape into two equal parts. There are three main types of bisectors:

1. Angle Bisector: An angle bisector is a line or ray that divides an angle into two equal parts. The point where the angle bisector meets the angle is called the vertex. Angle bisectors help in solving problems related to angles, such as finding the measure of an unknown angle or proving that two angles are equal.
2. Perpendicular Bisector: A perpendicular bisector is a line or plane that intersects a line segment at its midpoint and forms a right angle (90 degrees) with the segment. Perpendicular bisectors are useful for solving problems involving line segments, such as finding the distance between two points or proving that a quadrilateral is a parallelogram.
3. Median: A median in a triangle is a line that connects the midpoint of one side to the opposite vertex. Medians are used to solve problems related to triangles, like finding the centroid (the point where all medians intersect) or proving that a triangle is equilateral.

Line Segment Bisector

A line segment bisector is a line or ray that cuts a line segment exactly in half at its midpoint. This midpoint is the point that is the same distance from both ends of the line segment. The bisector is perpendicular to the line segment, meaning it forms a right angle (90 degrees) with the line segment at the midpoint.

Properties of a Line Segment Bisector

1. Perpendicular: The bisector is at a right angle to the line segment.
2. Equal Parts: It divides the line segment into two equal halves.
3. Midpoint: The bisector goes through the midpoint of the line segment.
4. Mirror Images: The two halves of the line segment are identical, or mirror images, on either side of the bisector.

What is a Perpendicular Bisector?

A perpendicular bisector is a line or ray that cuts a line segment exactly in half at its midpoint, forming a right angle (90 degrees) with the segment. This bisector creates two equal parts, making each part a mirror image of the other along the bisector.

Properties of Perpendicular Bisector

1. Passes Through the Midpoint: It intersects the line segment at its midpoint.
2. Right Angle: It forms a right angle (90 degrees) with the line segment.
3. Creates Two Equal Parts: It divides the line segment into two equal, congruent parts.
4. Mirror Images: The two halves of the line segment are mirror images of each other across the bisector.

Perpendicular Bisector Theorem

According to the Perpendicular Bisector Theorem, if a point is on the perpendicular bisector of a line segment, it is the same distance from both endpoints of the segment. This means that any point lying on the perpendicular bisector is equidistant from the segment's endpoints.

To prove:  If a point P lies on the perpendicular bisector of a segment AB, then AP = BP, where A and B are the endpoints of the segment AB.

Proof: To visualize this theorem, imagine a line segment AB with midpoint M and a point P that lies on the perpendicular bisector of AB. Then, triangles AMP and BMP are congruent by the Side-Angle-Side (SAS) congruence criterion, since AM = BM (both are half of AB, which is the definition of midpoint) and MP is common to both triangles. Therefore, by the Congruence of Corresponding Parts, AP = BP.

Perpendicular Bisector Equation

Consider a line segment PQ joining the points (x¹,y¹) and (x²,y²). Then the equation of perpendicular bisector: Equation of RM: y-y¹ + y22= -x2 - x1y2 - y1 (x-x1 + x22).

How to Construct a Perpendicular Bisector?

To create a perpendicular bisector for a line segment, follow these simple steps:

1. Draw Line Segment:

   • Use a ruler to draw the line segment AB.

2. Find the Midpoint:

   • Measure the length of AB and mark its midpoint M.

3. Set Compass Radius:

   • Place the compass point on M and adjust the radius to more than half the length of AB.

4. Draw Arcs:

   • With the compass on M, draw an arc above and another below the line segment. Ensure both arcs intersect the segment at points P and Q.

5. Draw the Perpendicular Bisector:

   • Use a ruler to draw a straight line passing through the intersecting points P and Q.

6. Verify the Bisector:

   • The line through P and Q is the perpendicular bisector of AB because it passes through M and is equidistant from points A and B.

The construction works because points P and Q, formed by the arcs, are equidistant from A and B. The perpendicular bisector goes through M, ensuring it is the midpoint of AB. Adjust the compass carefully to ensure proper intersection of the arcs with the segment.

Angle Bisector

An angle bisector is a line or segment that splits an angle into two equal parts. The vertex of the angle is where this bisector meets the angle.

Constructing an Angle Bisector:

1. Draw the angle using a ruler and protractor, placing the vertex at the center.
2. Place the compass point on the vertex and set it to a convenient length.
3. Draw arcs from the vertex to intersect each side of the angle, marking points A and B.
4. Draw a straight line through the vertex and the point where the arcs intersect.

This line is the angle bisector because the arcs' intersection is equidistant from the angle's sides, ensuring the angle is split into two equal parts.

Angle Bisector Theorem

The Angle Bisector Theorem states that an angle bisector in a triangle divides the opposite side into two segments proportional to the other two sides.

For triangle ABC, with the angle at vertex A bisected by a line meeting BC at point D, the theorem states: BDDC=ABAC\frac{BD}{DC} = \frac{AB}{AC}DCBD=ACAB

This means the segments BD and DC on side BC are in the same ratio as the lengths of sides AB and AC.

The theorem can be proved by showing that triangles ABD and ACD are similar, which means their corresponding sides are proportional. This leads to: BDAB=DCAC\frac{BD}{AB} = \frac{DC}{AC}ABBD=ACDC Cross-multiplying gives: BDDC=ABAC\frac{BD}{DC} = \frac{AB}{AC}DCBD=ACAB

These simplified steps and explanations help in understanding how angle bisectors work and their properties in geometry.