What do you call a line ray or segment which cuts a line segment into two equal parts at 90?

All good learning begins with vocabulary, so we will focus on the two important words of the theorem. Perpendicular means two line segments, rays, lines or any combination of those that meet at right angles. A line is perpendicular if it intersects another line and creates right angles.

Bisector

A bisector is an object (a line, a ray, or line segment) that cuts another object (an angle, a line segment) into two equal parts. A bisector cannot bisect a line, because by definition a line is infinite.

Perpendicular Bisector

Putting the two meanings together, we get the concept of a perpendicular bisector, a line, ray or line segment that bisects an angle or line segment at a right angle.

Before you get all bothered about it being a perpendicular bisector of an angle, consider: what is the measure of a straight angle? 180°; that means a line dividing that angle into two equal parts and forming two right angles is a perpendicular bisector of the angle.

Perpendicular Bisector Theorem

Okay, we laid the groundwork. So putting everything together, what does the Perpendicular Bisector Theorem say?

If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment.

How Does It Work?

Suppose you have a big, square plot of land, 1,000 meters on a side. You built a humdinger of a radio tower, 300 meters high, right smack in the middle of your land. You plan to broadcast rock music day and night.

Anyway, that location for your radio tower means you have 500 meters of land to the left, and 500 meters of land to the right. Your radio tower is a perpendicular bisector of the length of your land.

You need to reinforce the tower with wires to keep it from tipping over in high winds. Those are called guy wires. How long should a guy wire from the top down to the land be, on each side?

Because you constructed a perpendicular bisector, you do not need to measure on each side. One measurement, which you can calculate using geometry, is enough. Use the Pythagorean Theorem for right triangles:

Your tower is 300 meters. You can go out 500 meters to anchor the wire's end. The tower meets your land at 90°. So:

300 m2 + 500 m2 = c2

90,000 + 250,000 = c2

340,000 = c2

340,000 = c

583.095 m = c

You need guy wires a whopping 583.095 meters long to run from the top of the tower to the edge of your land. You repeat the operation at the 200 meter height, and the 100 meter height.

For every height you choose, you will cut guy wires of identical lengths for the left and right side of your radio tower, because the tower is the perpendicular bisector of your land.

Proving the Perpendicular Bisector Theorem

Behold the awesome power of the two words, "perpendicular bisector," because with only a line segment, HM, and its perpendicular bisector, WA, we can prove this theorem.

We are given line segment HM and we have bisected it (divided it exactly in two) by a line WA. That line bisected HM at 90° because it is a given. This means, if we run a line segment from Point W to Point H, we can create right triangle WHA, and another line segment WM creates right triangle WAM.

What do we have now? We have two right triangles, WHA and WAM, sharing side WA, with all these congruences:

1. WA ≅ WA (by the reflexive property)
2. ∠WAH ≅ ∠WAM (90° angles; given)
3. HA ≅ AM (bisector; given)

What does that look like? We hope you said Side Angle Side, because that is exactly what it is.

The Side Angle Side Postulate states, "If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent."

That means sides WH and WM are congruent, because CPCTC (corresponding parts of congruent triangles are congruent). WHAM! Proven!

Practice Proof

You can tackle the theorem yourself now. You will either sink or swim on this one. Here is a line segment, WM. We construct a perpendicular bisector, SI.

How can you prove that SW ≅ SM? Do you know what to do?

1. Construct line segments SW and SM.
2. You now have what? Two right triangles, SWI and SIM. They have right angles, ∠SIW and ∠SIM.
3. Identify WI and IM as congruent, because they are the two parts of line segment WM that were bisected by SI.
4. Identify SI as congruent to itself (by the reflexive property).

What does that give you? Two congruent sides and an included angle, which is what postulate? The SAS Postulate, of course! Therefore, line segment SW ≅ SM.

So, did you sink or SWIM?

Converse of the Perpendicular Bisector Theorem

Notice that the theorem is constructed as an "if, then" statement. That immediately suggests you can write the converse of it, by switching the parts:

If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment.

We can show this, too. Construct a line segment HD. Place a random point above it (but still somewhere between Points H and D) and call it Point T. If Point T is the same distance from Points H and D, this converse statement says it must lie on the perpendicular bisector of HD.

You can prove or disprove this by dropping a perpendicular line from Point T through line segment HD. Where your perpendicular line crosses HD, call it Point U.

If Point T is the same distance from Points H and D, then HU ≅ UD. If Point T is not the same distance from Points H and D, then HU ≇ UD.

The symbol ≇ means "not congruent to."

You can go through the steps of creating two right triangles, △THU and △TUD and proving angles and sides congruent (or not congruent), the same as with the original theorem.

You would identify the right angles, the congruent sides along the original line segment HD, and the reflexive congruent side TU. When you got to a pair of corresponding sides that were not congruent, then you would know Point T was not on the perpendicular bisector.

Only points lying on the perpendicular bisector will be equidistant from the endpoints of the line segment. Everything else lands with a THUD.

Lesson Summary

After you worked your way through all the angles, proofs and multimedia, you are now able to recall the Perpendicular Bisector Theorem and test the converse of the Theorem. You also got a refresher in what "perpendicular," "bisector," and "converse" mean.

Next Lesson:

Congruency of Right Triangles

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.

1. 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.

2. 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.

3. 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.