Geometry

6.2 & 6.3 Triangle Concurrency Points

Essential Question

What are points of concurrency?

January 14, 2016 6.2 and 6.3 Points of Concurrency

January 14, 2016 6.2 and 6.3 Points of Concurrency

Goals

Review terms.

Know what medians and altitudes are.

Review characteristics of each concurrent

point.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Concurrent Lines

Three or more lines intersecting at the same

point are concurrent.

The point where they intersect is the point of

concurrency.

Perpendicular Bisector

The perpendicular bisector of a line segment

is the line that is perpendicular to the

segment at its midpoint.

January 14, 2016 6.2 and 6.3 Points of Concurrency

A B

R

S

Angle Bisector

The angle bisector is a ray that divides an

angle into two congruent adjacent angles.

January 14, 2016 6.2 and 6.3 Points of Concurrency

DA

B

C

January 14, 2016 6.2 and 6.3 Points of Concurrency

Altitude

An altitude of a triangle is a segment drawn

from a vertex perpendicular to the opposite

side (or to the line containing the opposite

side).

A triangle has three altitudes.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Median

A median of a

triangle is the

segment

drawn from a

vertex to the

midpoint of the

opposite side.

Points of Concurrency

Four points of concurrency:

Circumcenter

Incenter

Centroid

Orthocenter

Made using different types of lines and have

a variety of properties.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Created when using the perpendicular

bisectors of each side of a triangle.

In the example box, draw one of the

perpendicular bisectors of the triangle.

Concurrent Point: Circumcenter

January 14, 2016 6.2 and 6.3 Points of Concurrency

When all three perpendicular bisectors are

drawn, the point of concurrency created is

called the circumcenter.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Concurrent Point: Circumcenter

January 14, 2016 6.2 and 6.3 Points of Concurrency

Circumcenter Property

The circumcenter is equidistant from each

vertex of the triangle.This is called a

circumcircle.

Created when using the angle bisectors of each

vertex of a triangle.

In the example box, draw one of the angle

bisectors of the triangle.

Concurrent Point: Incenter

January 14, 2016 6.2 and 6.3 Points of Concurrency

January 14, 2016 6.2 and 6.3 Points of Concurrency

Concurrent Point: Incenter

When all three angle bisectors are drawn, the

point of concurrency created is called the

incenter.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Incenter Property

The incenter is equidistant from the sides of a

triangle.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Incenter Property

The incenter is equidistant from the sides of a

triangle.

This is called

an incircle.

Created when using the medians of a triangle.

In the example box, draw one of the medians of

the triangle.

Concurrent Point: Centroid

January 14, 2016 6.2 and 6.3 Points of Concurrency

January 14, 2016 6.2 and 6.3 Points of Concurrency

Concurrent Point: Centroid

When all three medians are drawn, the point

of concurrency created is called the centroid.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Centroid Property

The centroid of a triangle is two thirds of the distance from each vertex to the midpoint of the opposite side. A

B

C

23

13

AB AC

BC AC

January 14, 2016 6.2 and 6.3 Points of Concurrency

Centroid Property

x2

3x

1

3x

January 14, 2016 6.2 and 6.3 Points of Concurrency

Examples

Turn to your notes to practice a few

examples.

C is the centroid of the triangle.

C

S

R

January 14, 2016 6.2 and 6.3 Points of Concurrency

Example 1

C

S

RRS = 9

RC = ?

CS = ? 6

3

January 14, 2016 6.2 and 6.3 Points of Concurrency

Example 2

C

S

RRS = 15

RC = ?

CS = ? 10

5

January 14, 2016 6.2 and 6.3 Points of Concurrency

Example 3

C

S

RRS = ?

RC = 8

CS = ? 12

4

8

January 14, 2016 6.2 and 6.3 Points of Concurrency

Example 4

C

S

RRS = ?

RC = 30

CS = ? 45

15

30

January 14, 2016 6.2 and 6.3 Points of Concurrency

Example 5

C

S

RRS = ?

RC = ?

CS = 7 21

7

14

January 14, 2016 6.2 and 6.3 Points of Concurrency

Example 6

C

S

RRS = ?

RC = ?

CS = 4.2 12.6

4.2

8.4

January 14, 2016 6.2 and 6.3 Points of Concurrency

Centroid Property

The centroid of a

triangle is also known

as the center of

balance.

Turn back to the graphic organizer.

Created when using the altitudes of a triangle.

In the example box, draw one of the altitudes of

the triangle.

Concurrent Point: Orthocenter

January 14, 2016 6.2 and 6.3 Points of Concurrency

January 14, 2016 6.2 and 6.3 Points of Concurrency

Concurrency Point: Orthocenter

When all three altitudes are drawn, the point of

concurrency created is called the orthocenter.

January 14, 2016 6.2 and 6.3 Points of Concurrency

Orthocenter Property

None!

January 14, 2016 6.2 and 6.3 Points of Concurrency

Fast answers!

The altitudes are concurrent at the ?

Orthocenter

The medians are concurrent at the ?

Centroid

The perpendicular bisectors are concurrent at the ?

Circumcenter

The angle bisectors are concurrent at the ?

Incenter

January 14, 2016 6.2 and 6.3 Points of Concurrency

Fast answers!

Which point is equidistant from the sides of a

triangle?

Incenter

Which point is the center of balance?

Centroid

Which point is equidistant from the vertices?

Circumcenter?

January 14, 2016 6.2 and 6.3 Points of Concurrency

Fast answers!

What point is needed to draw a circumcircle?

Circumcenter

What point is needed to draw an incircle?

Incenter

What point is needed to find the center of

balance?

Centroid

January 14, 2016 6.2 and 6.3 Points of Concurrency

Do you know…

What a median is?

How to draw a perpendicular bisector?

What lines are needed to find the incenter?

How to locate the circumcenter?

What point is located using the medians?

How to construct an altitude?

Which concurrent point is the same distance

from each vertex of a triangle?