Triangles: Points of Concurrency

MM1G3 e

Investigate Points of Concurrency

• http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html

Circumcenter

Perpendicular Bisectors and Circumcenters Examples

A perpendicular bisector of a triangle is a line or line segment that forms a right angle with one side of the triangle at the midpoint of that side. In other words, the line or line segment will be both perpendicular to a side as well as a bisector of the side.

DCB

A

A

BCD

E

F

it will. sometimes however,

a vertex; throughgo obisector tlar perpendicua for necessary

not isIt vertices. theofany containnot does that Notice*

. ofbisector lar perpendicua is

EF

ABCEF

. that Notice

side.a bisects also only However,

. of side one lar toperpendicu are and Both

BFAF

EF

ABCEFAD

Example 1:

Example 2:

Q

P N

M

€

In ΔMNP above, MQ ≅ NQ and PQ⊥MN .

Therefore, PQ is the perpendicular bisector of MN .

*Notice that PQ contains the vertex P.

Since a triangle has three sides, it will have three perpendicular bisectors. These perpendicular bisectors will meet at a common point – the circumcenter.

FE

G

D

G is the circumcenter of ∆DEF.

Notice that the vertices of the triangle (D, E, and F) are also points on the circle. The circumcenter, G, is equidistant to the vertices.

. So, FGEGDG

The circumcenter will be located inside an acute triangle (fig.1), outside an obtuse triangle (fig. 2), and on a right triangle (fig. 3). In the triangles below, all lines are perpendicular bisectors. The red dots indicate the circumcenters.

fig. 1 fig. 3

fig. 2

Example 3: A company plans to build a new distribution center that is convenient to three of its major clients, as shown below. Why would placing this distribution center at the circumcenter be a good idea?

The circumcenter is equidistant to all three vertices of a triangle. If the distribution center is built at the circumcenter, C, the time spent delivering goods to the three major clients would be the same.

C

In Summary

• The circumcenter is the point where the three perpendicular bisectors of a triangle intersect.

• The circumcenter can be inside, outside, or on the triangle.

• The circumcenter is equidistant from the vertices of the triangle

Circumcenter

• Exploration

• Construction

Try These:

C

Try These:

D

Try These:

A

Try These

B

Medians and Centroids Examples

A median of a triangle is a line segment that contains the vertex of the triangle and the midpoint of the opposite side. Therefore, the median bisects the side.

triangle. theof mediana is .

and ofmidpoint theis Therefore, . bisects above, In

ADCDBD

BCDBCADABC

Since a triangle has three sides, it will have three medians. These medians will meet at a common point – the centroid.

. of centroid theis

.point at intersect medians

The triangle. theof medians are

and , , that see can we

, on markings theFrom

ABCO

O

CFBEAD

ABC

The centroid is always located inside the triangle.

Acute triangle

The distance from any vertex to the centroid is 2/3 the length of the median. Q

RSD

E FG

8

1232

32

.12 Suppose . of length the

of 32 is centroid the to vertex from

distance The median.a is , In

QG

QG

QDQG

QDQD

GQ

QDQRS

Example 1: G is the centroid of triangle QRS. QG = 10 GF = 3. Find QD and SF.

Q

RSD

E FG

QD

QD

QDQG

153

210

32

SF

SF

SFGFSFSG

9

3

. then, Since

31

31

32

Example 3: G is the centroid of triangle DEF. FG = 15, ES = 21, QG = 5 Determine FR, EG and GD

Q

R

S

D

E

F G

FR

FR

FRFG

5.22

15 32

32

15

21

5

14

2132

32

EG

EG

ESEG

10

52

.2 then

,

and Since

31

32

GD

GD

QGGD

QDQG

QDGD

Notice that the distance from any vertex to the centroid is 2/3 the length of the median. That means that the distance from the centroid to the midpoint of the opposite side is 1/3 the length of the median.

So, in triangle MNP, MQ=2(QT) and QT=(1/2)MQ

Q

P

N

U

M

T

V

The centroid is also known as the balancing point (center of gravity) of a triangle.

In Summary• A median is a line segment from the a vertex of a triangle

to the midpoint of the opposite side.

• The distance from the vertex to the centroid is 2/3 the length of the median.

• The distance from the centroid to the midpoint is 1/3 the length of the median, or half the distance from the vertex to the centroid.

• Since the centroid is the balancing point of the triangle, any triangular item that is hung by its centroid will balance.

Centroids

• Investigate

• Construction

Try These:

D

Try These:

C

Try These:

A

Try These:

C

Angle Bisectors and Incenters Examples

An angle bisector of a triangle is a segment that shares a common endpoint with an angle and divides the angle into two equal parts.

triangle. theofbisector angle an is Therefore

. so parts equal twointo divides , In

AD

CADBADBACADABC

Example 1: Determine any angle bisectors of triangle ABC.

A

BC

D

E

FG

triangle. theof

bisector angle an is Therefore,

. that see can we

triangle, theon markings theFrom

BG

CBGABG

Since a triangle has three angles, it will have three angle bisectors. These angle bisectors will meet at a common point – the incenter.

M

ZY

X

triangle. theofincenter theis Therefore,

.point at intersect bisectors angle three the, In

M

MXYZ

The incenter is always located inside the triangle.

Acute triangle

Right triangle

Obtuse triangle

incenter

The incenter is equidistant to the sides of the triangle.

triangle. theof sides theof each

from , distance, same theis

incenter. theis , on,intersecti

ofpoint Thebisectors. angle are

and ,,, In

xG

G

CFBEADABC

x

x

Example 2: L is the incenter of triangle ABC. Which segments are congruent?

L

A

B

C

D

EF

. Therefore,

triangle. theof sides the topoint

from distance therepresent and

,, segments,lar perpendicu

The triangle. theof sides the

t toequidistan isincenter The

LFLELD

L

LF

LELD

Example 3: Given P is the incenter of triangle RST. PN = 10 and MT = 12, find PM and PT.

Not drawn to scale

10

12

Theorem. an Pythagoreby the

find can wengle,right triaa is

since Also, .10

So, .triangle,

theofincenter theis Since

PT

PMTPM

PMPN

P

PT

PT

PT

PTMTPM

612

244

1210

.

2

222

222

In Summary

• The incenter is the point of intersection of the three angle bisectors of a triangle.

• The incenter is equidistant to all three sides of the triangle.

Incenter

• Investigate

• Construction

Try These:

B

Try These:

C

Try These:

C

Try These:

K C

Altitudes and Orthocenters Examples

An altitude of a triangle is the perpendicular distance between a vertex and the opposite side. This distance is also known as the height of the triangle.

D C

B

A

. vertex thecontains

and lar toperpendicu is that Notice . of altitude an is

A

BCADABCAD

Example 1: Determine any altitudes of triangle ABC.

FE

D

C

BA

. side, opposite thelar toperpendicunot isit since altitude annot is

. vertex, thecontains and lar toperpendicu is . vertex, thecontains

and lar toperpendicu is altitudes. are and only , In

ACBE

CABCDA

BCAFCDAFABC

Since a triangle has three sides, it will have three altitudes. These altitudes will meet at a common point – the orthocenter.

O

Z

Y

X

. ofr orthocente theis

. point, common at themeet altitudes These

triangle. theof altitude an each are and ,, that Notice

ABCO

O

CXBYAZ

The orthocenter may be located inside an acute triangle (fig. 1), outside an obtuse triangle (fig. 2), or on a right triangle (fig. 3). In the triangles below, the red lines represent altitudes. The red points indicate the orthocenters.

Obtuse TriangleAcute Triangle Right Triangle

Fig. 3Fig. 2Fig. 1

Summary• An altitude is a line segment containing a

vertex of a triangle and is perpendicular to the opposite side.

• The orthocenter is the intersection point of the three altitudes of a triangle.

• Orthocenters can be inside, outside, or on the triangle depending on the type of triangle.

Orthocenter

• Investigate

• Construction

Try These:

B

Try These:

B

Try These:

A

Points of Concurrency

Investigate

Points of Concurrency

MM1G3 eReview

Try These:

A

Try These:

B

Try These:

D

Try These:

B

Try These:

A

Try These:

A

Try These: B

Try These:

D

Try These:

A

Try These:

C