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This pdf includes the following topics:-

Midpoint

Segment bisector

The Midpoint Formula

Bisecting an Angle

Midpoint

Segment bisector

The Midpoint Formula

Bisecting an Angle

1.
Chapter 1

Basics of Geometry

Basics of Geometry

2.
Section 5

Segment and Angle Bisectors

Segment and Angle Bisectors

3.
The __midpoint__ of a segment is the point that divides, or __bisects__, the

segment into two congruent segments. In this book, matched 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.

C

A M B A M B

D

M is the midpoint of AB if

M is on AB and AM = MB. CD is a bisector of AB.

segment into two congruent segments. In this book, matched 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.

C

A M B A M B

D

M is the midpoint of AB if

M is on AB and AM = MB. CD is a bisector of AB.

4.
You can use a compass and a straightedge (a ruler without marks) to

construct 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.

construct 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.

5.
If you know the coordinates of the endpoints of a segment, you can calculate

the coordinates of the midpoint. You simply take the mean, or average, of the

x-coordinates and of the y-coordinates. This methods is summarized as the

Midpoint Formula.

the coordinates of the midpoint. You simply take the mean, or average, of the

x-coordinates and of the y-coordinates. This methods is summarized as the

Midpoint Formula.

6.
The Midpoint Formula

If A(x1, y1) and B(x2, y2) are points

in a coordinate plane, then the

midpoint of AB has coordinates

x1 x 2 y1 y 2

,

2 2

If A(x1, y1) and B(x2, y2) are points

in a coordinate plane, then the

midpoint of AB has coordinates

x1 x 2 y1 y 2

,

2 2

7.
Example 1: Finding the Coordinates of the Midpoint of a Segment

Find the coordinates of the midpoint of AB with endpoints

A(-2, 3) and B(5, -2).

Find the coordinates of the midpoint of AB with endpoints

A(-2, 3) and B(5, -2).

8.
Example 2: Finding the Coordinates of an Endpoint of a Segment

***TEST***

The midpoint of RP is M(2, 4). One endpoint is R(-1, 7). Find the coordinates of

the other endpoint.

***TEST***

The midpoint of RP is M(2, 4). One endpoint is R(-1, 7). Find the coordinates of

the other endpoint.

9.
GOAL 2: Bisecting an Angle

A

An angle bisector is a ray that divides an

angle into two adjacent angles that are

D

congruent. In the diagram at the right, the C

ray CD bisectsangle into two congruent angles, and In this book, matching congruence arcs

identify congruent angles in diagrams.

A

An angle bisector is a ray that divides an

angle into two adjacent angles that are

D

congruent. In the diagram at the right, the C

ray CD bisects

identify congruent angles in diagrams.

10.
Example 3: Dividing an Angle Measure in Half

The ray FH bisects the anglem H

F

120°

120/2 = 60

m m

The ray FH bisects the angle

F

120°

120/2 = 60

m

11.
Example 4: Doubling an Angle Measure

K

In the kite, two angles are bisected. 45°

E I

Find the measure of the two angles.

m m

K

In the kite, two angles are bisected. 45°

m

12.
Example 5: Finding the Measure of an Angle

In the diagram, RQ bisectof the two congruent angles are (x + 40)° and

(3x – 20)°. Solve for x.

(x + 40)° Q

R

(3x – 20)°

S

In the diagram, RQ bisect

(3x – 20)°. Solve for x.

(x + 40)° Q

R

(3x – 20)°

S