A perpendicular bisector is a line that cuts a segment in half (through the midpoint) and is perpendicular to the segment that's being cut.
A segment bisector is a line, a ray, a line segment, or a point that cuts a line segment at the center dividing the line into two equal parts. The word segment can also be referred to as line segment that means a segment is a part of the line that has fixed endpoints. The word bisect means cutting any object or line into two equal halves. Hence, a segment bisector is referred to as when two line segments bisect or cut each other at a point dividing the lines into equal halves. Let us learn more about segment bisector and solve a few examples.
Definition of Segment Bisector
Segment bisector is a line, ray, or segment that cuts another line segment at the center dividing the line into two equal halves. The line always bisects or passes through the midpoint of the line segment dividing it into two equal parts. The midpoint can have one or infinite segments bisecting the line and not necessarily be only a perpendicular bisector. Let us look at the image given below.
Line AB is divided into two equal halves i.e. AM and MB by the segment bisector XY. If the line XY cuts the line segment at exactly 90°, it is said to be a perpendicular bisector. But in this case, the line does not cut at a right angle, hence it is a segment bisector. The point M is considered to be the midpoint of the line segment AB where AM = MB.
Types of Segment Bisector
A segment bisector divides a line into two or more equal parts. There are different types of segment bisectors that define bisecting a line. They are:
- Points
- Lines
- Ray
- Line Segment
Points
A point is defined as a location in any space or object represented by a dot (.). It does not have any length, height, shape, or size but when two points are connected they make a line. Hence, a point marks the beginning to draw any figure or shape and is written with capital letters. Two or more points that lie on a single straight line are collinear points. Two or more points that lie on the same plane are coplanar points. In a segment bisector, a point helps an essential role as it marks the point on the line that divided the line into two halves and it is called as the midpoint. Only a line segment can have a midpoint and not a line or ray.
Lines
A line is a figure when two points are connected with a distance between them along with ends extending to infinite. In other words, a line is a straight path constructed by a series of points. A line has no thickness and can extend indefinitely in both directions. The length of a line is undefined and it can have infinite numbers of points. There are different types of lines we learn in geometry such as parallel lines, perpendicular lines, horizontal lines, intersecting lines, and vertical lines. Parallel lines do not intersect each other while lines that intersect at 90° are called perpendicular lines.
Ray
A ray is a part of a line that has only one fixed point and the other point does not have any end. While rays have a fixed beginning and no definite end, they are represented in our day-to-day lives with examples such as the sunlight or the light of a torch. A ray is represented with a small arrow above the points. For example in the ray seen below, we can write it as \[\overrightarrow{\rm AB}\]. Where A is the endpoint while B is the point through which the ray is extended.
Line Segment
Line segment is the path between two points that can be measured. Since line segments have a defined length, they can form the sides of any polygon. The figure given below shows a line segment AB, where the length of line segment AB refers to the distance between its endpoints, A and B.
Segment Bisector and Midpoints
In a segment bisector, the point that bisects the lines into two equal halves is called a midpoint. By definition, a midpoint is a point lying in the middle or center of a line joining the two points. For the two points, if a line is drawn joining the two points, then the midpoint is a point at the middle of the line and is equidistant from the two points. Through this midpoint there could be a ray or a line passing by that divides the line into equal parts. Multiply rays or line segments can also pass by the same midpoint as the segment bisector. To determine if the line segment is a segment bisector, we can verify if it crosses on the midpoint and if it does pass on the midpoint, we can use the midpoint formula to find the coordinates of the line. The midpoint formula is:
[(x1 + x2)/2, (y1 + y2)/2)]
Where,
- (x1 + x2)/2 is average of x-coordinates.
- (y1 + y2)/2 is average of y-coordinates.
Segment Bisectors and Perpendicular Bisectors
Segment bisectors that bisect at 90° are called perpendicular bisectors. A perpendicular bisector is defined as a line or a line segment that divides a given line segment into two parts of equal measurement making four angles of 90° each on both sides. Perpendicular bisector on a line segment can be constructed easily using a ruler and a compass.
Related Topics
Listed below are a few interesting topics related to segment bisector, take a look.
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Example 1: Find at which point a perpendicular bisector bisects a line segment of length 20 units.
Solution:
A perpendicular bisector is a line that bisects a given line segment into two congruent line segments exactly at its midpoint. It is given that the line segment is of the length of 20 units. So, the perpendicular bisector bisects the line segment exactly at 10 units and the line segment of 20 units is divided into two line segments of 10 units each. -
Example 2: Consider the line segment \(\overline{AB}\). The endpoints are (3, h) and (7, 7). Find the value of h if the midpoint of \(\overline{AB}\) is (4, -2).
Solution:
Let x1 = 3, y1 = h, x2 = 7, and y2 = 7. According to the definition of midpoint we have, (x1 + x2)/2, (y1 + y2)/2) = ((3 + 7)/2, (h + 7)/2) = (10/2, (h + 7)/2) = (5, (h + 7)/2). Equalizing this with the midpoint value (4, -2) we have (h + 7)/2 = -2; h + 7 = -2 × 2; h + 7 = -4; h = -4 - 7; h = -11. Therefore, the value of h is -11.
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Example 3: Identify if the given figure is a line segment, a line, or a ray.
Solution:
The figure has one starting point but an arrow on the other end. This shows that it is not a line segment or a line, it is a ray. Therefore, PQ is a ray.
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FAQs on Segment Bisector
A segment bisector is a line or ray or line segment that passes through the midpoint of another line segment dividing the line into two equal parts.
How Do You Find the Segment Bisector?
To find the segment bisector of a line segment, we use the midpoint formula. [(x1 + x2)/2, (y1 + y2)/2)].
Where,
- (x1 + x2)/2 is average of x-coordinates.
- (y1 + y2)/2 is average of y-coordinates.
What is the Difference Between Segment Bisector and Perpendicular Bisector?
A segment bisector is any line, ray, or segment that divides another segment into two equal parts. Whereas a perpendicular bisector is a form of segment bisector that divides another segment into equal parts along with forming a 90-degree angle.
Does a Segment Bisector Have to be Perpendicular?
A segment bisector can be perpendicular when the line segment is at 90°. But a segment bisector can also be at other angles irrespective of the direction. Hence, a segment bisector doesn't always have to be perpendicular.
What are the Types of Segment Bisectors?
A segment bisector is of 4 different types, namely points, lines, line segment, and ray. Each of these can bisect the other at the midpoint making them segment bisectors.