# SEGMENT AND ANGLE BISECTORS

## About "Segment and angle bisectors"

Segment and angle bisectors :

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

An angle bisector is a line or ray that divides an angle into two congruent angles. There are two types of angle bisectors. They are interior and exterior.

## Bisecting a segment

The midpoint of a segment is the point that divides, or the segment into two congruent segments. In this book, matching red congruence marks identify congruent segments in diagrams.

A segment bisector is a segment, ray, line, or plane that intersects a segment at its midpoint.

M is the mid point AB. Because M is on AB, we have

AM  =  MB.

CD is a bisector of AB.

We can use a compass and a straightedge (a ruler without marks) to a segment bisector and midpoint of AB. A construction is a geometric drawing that uses a limited set of tools, usually a compass and a straightedge.

## Segment Bisector and Midpoint - Construction

Use the following steps to construct a bisector of AB and find the midpoint M of AB.

Step 1 :

Place the compass point at A. Use a compass setting greater than half the length of AB.

Draw an arc.

Step 2 :

Keep the same compass setting. Place the compass point at B. Draw an arc. It should intersect the other arc in two places.

Place the compass point at A. Use a compass setting greater than half the length of AB.

Draw an arc.

Step 3 :

Use a straightedge to draw a segment through the points of intersection. This segment bisects AB at M, the midpoint of AB. Keep the same compass setting. Place the compass point at B. Draw an arc. It should intersect the other arc in two places. Place the compass point at A. Use a compass setting greater than half the length of AB.

Draw an arc.

## The Midpoint Formula

If we know the coordinates of the endpoints of a segment, we can calculate the coordinates of the midpoint. We simply take the mean, or average, of the x-coordinates and of the y-coordinates. This method is summarized as the Midpoint formula.

Let A(x, y) and B(x, y) be the two points in a coordinate plane as shown below.

Then the midpoint of AB has coordinates

## Bisecting an Angle

An angle bisector is a ray that divides an angle into two adjacent angles that are congruent.

In the diagram given above, the ray CD bisects ABC, because it divides the angle into two congruent angles, ACD and BCD.

## Angle Bisector - Construction

Use the following steps to construct an angle bisector of C.

Step 1 :

Place the compass point at C. Draw an arc that intersects both sides of the angle. Label the intersections A and B.

Step 2 :

Place the compass point at A. Draw an arc. Then place the compass point at B. Using the same compass setting, draw another arc.

Step 3 :

Label the intersection D. Use a straightedge to draw a ray through C and D. This is the angle bisector. Place the compass point at A. Draw an arc. Then place the compass point at B. Using the same compass setting, draw another arc.

After we have constructed an angle bisector, we should check that it divides the original angle into two congruent angles. One way to do this is to use a protractor to check that the angles have the same measure.

Another way is to fold the piece of paper along the angle bisector. When we hold the paper up to a light, we should be able to see that the sides of the two angles line up, which implies that the angles are congruent.

Fold on CD.

The sides of angles BCD and ∠ACD line up.

After having gone through the stuff given above, we hope that the students would have understood "Segment and angle bisectors".

Apart from the stuff "Segment and angle bisectors", if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6