Bisector

Apr 29, 2023, 16:45 IST

Bisectors are geometric lines or planes that divide an angle, line segment or shape into two equal parts. They are essential tools in many areas of mathematics, such as geometry, trigonometry, and calculus. Bisectors can help solve problems related to angles, triangles, circles, and other geometric shapes.

Introduction

Bisectors are one of the fundamental concepts in geometry. They can be defined as a line, ray, or plane that divides an angle, line segment, or shape into two equal parts. The word "bisect" comes from the Latin word "bisector," which means "to cut into two equal parts."

Bisectors play a crucial role in geometry, as they allow us to solve problems related to angles, triangles, and circles. They can also help us find the center of mass of a shape, which is important 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:

• Angle Bisector - An angle bisector is a line or ray that divides an angle into two equal parts. The point where the angle bisector intersects the angle is called the vertex of the angle. Angle bisectors are used to solve problems related to angles, such as finding the measure of an unknown angle or proving that two angles are equal.
• Perpendicular Bisector - A perpendicular bisector is a line or plane that intersects a line segment at its midpoint and forms a right angle with the line segment. Perpendicular bisectors are used to solve problems related to line segments, such as finding the distance between two points or proving that a quadrilateral is a parallelogram.
• Median - A median is a line or plane that connects the midpoint of a line segment to the vertex of an opposite angle in a triangle. Medians are used to solve problems related to triangles, such as finding the centroid of a triangle or proving that a triangle is equilateral.

Line Segment Bisector

A line segment bisector is a line or ray that intersects a line segment at its midpoint and divides it into two equal parts. The midpoint of the line segment is the point that is equidistant from the two endpoints of the line segment. The line segment bisector is perpendicular to the line segment at its midpoint, forming two right angles.

Properties of Line Segment Bisector

• A line segment bisector is perpendicular to the line segment.
• A line segment bisector divides the line segment into two congruent parts.
• It passes through the midpoint of the line segment.
• The two parts of the line segment are mirror images of each other with respect to the line segment bisector.

What is a Perpendicular Bisector?

A perpendicular bisector is a line or a ray that intersects a line segment at its midpoint and forms a right angle with it. It divides the line segment into two congruent parts, and each part is a mirror image of the other with respect to the perpendicular bisector.

Properties of Perpendicular Bisector

• A perpendicular bisector passes through the midpoint of a line segment.
• A perpendicular bisector is perpendicular to the line segment.
• A perpendicular bisector divides the line segment into two congruent parts.
• The two parts of the line segment are mirror images of each other with respect to the perpendicular bisector.

For example, a perpendicular bisector can be used to find the center of a circle that passes through three non-collinear points. By finding the perpendicular bisectors of the three sides of the triangle formed by the three points, the center of the circle can be found as the point where the three perpendicular bisectors intersect.

Perpendicular Bisector Theorem

According to the Perpendicular Bisector Theorem, a point is equidistant from a segment's endpoints if it lies on the segment's perpendicular bisector. In other words, the distance from the point to each endpoint of the segment is the same.

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.

PerpPerpendicular Bisector Equation

Consider a line segment PQ joining the points (x1, y1) and (x2, y2).

Then the equation of perpendicular bisector:

Equation of RM:

How to Construct a Perpendicular Bisector?

To construct a perpendicular bisector of a line segment, we can follow the steps below:

• Draw the given line segment AB using a ruler.
• Locate the midpoint M of the line segment AB by measuring and marking the half-length of AB.
• Place the point of the compass on the midpoint M and adjust the radius to more than half the length of AB.
• Draw two arcs of the same radius, one above point A and one below point B, intersecting the line segment at points P and Q, respectively.
• Draw a straight line passing through points P and Q using a ruler.
• The line passing through points P and Q is the perpendicular bisector of the line segment AB.
• The construction works because the points P and Q are equidistant from the endpoints A and B of the line segment AB. The perpendicular bisector passes through the midpoint M, which is the only point that is equidistant from A and B.

It is essential to adjust the radius of the compass carefully to ensure that the two arcs intersect the line segment. If the radius is too small or too large, the arcs may not intersect at all or may intersect outside the line segment.

Angle Bisector

An angle bisector is a line or line segment that divides an angle into two equal parts. The vertex of the angle is the point where the angle's bisector intersects it.

To construct an angle bisector, we can follow the steps below:

• Draw the angle using a ruler and a protractor, with the vertex at the center of the protractor.
• Place the point of the compass on the vertex of the angle and adjust the radius to any convenient length.
• Draw two arcs, one on each ray of the angle, intersecting the rays at points A and B, respectively.
• Draw a straight line passing through the vertex of the angle and the point of intersection of the arcs.
• The line passing through the vertex and the point of intersection of the arcs is the angle bisector.

The construction works because the point of intersection of the arcs is equidistant from the sides of the angle, so it lies on the angle bisector. Moreover, the angle bisector divides the angle into two congruent angles, which have the same measure.

Angle Bisector Theorem

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

More precisely, suppose we have a triangle ABC, where the angle at vertex A is bisected by a line that intersects side BC at point D. Then, according to the Angle Bisector Theorem, we have:

BD/DC = AB/AC

In other words, the ratio of the lengths of the two segments into which the bisector divides side BC is equal to the ratio of the lengths of the two adjacent sides of the triangle.

The Angle Bisector Theorem can be proved using similar triangles. Specifically, we can show that triangle ABD and triangle ACD are similar, which implies that their corresponding sides are proportional. Therefore, we have:

BD/AB = DC/AC

Cross-multiplying and simplifying, we obtain:

BD/DC = AB/AC

Frequently Asked Questions (FAQs)

Q1. What is a bisector?

Ans. A bisector is a line or a plane that divides a given object or shape into two equal parts. In geometry, bisectors are often used to divide angles or line segments into two equal parts.

Q2. What is an angle bisector?

Ans. An angle bisector is a line or a ray that divides an angle into two equal angles. The point where the angle bisector intersects the angle is called the vertex of the angle.

Q3. What is a perpendicular bisector?

Ans. A perpendicular bisector is a line or a plane that intersects a line segment at its midpoint and is perpendicular to the line segment. A perpendicular bisector divides a line segment into two equal parts.

Q4. How do you find the length of a bisector?

Ans. To find the length of a bisector, you need to know the length of the object or shape that the bisector is dividing and the angle at which the bisector intersects it. Using trigonometry, you can calculate the length of the bisector based on these measurements.

Q5. What is the importance of bisectors in geometry?

Ans. Bisectors are important in geometry because they help us to divide angles and line segments into equal parts. This is useful in many different contexts, such as when constructing shapes or when calculating distances and angles in real-world applications. Bisectors are also used in proofs and theorems in geometry.