Congruent Triangles

4.52 Importance of concurrency

TrianglePoints of Concurrency

#1Perpendicular bisectors

and the circumcenter.

C

T

A

B

3PerpBisectors

Before we can talk about the circumcenter’s importance, we need some review on perpendicular bisectors.

Circumcenter

BA

M BA

Review of Perpendicular Bisector Properties.

M BA

Every point on the perpendicular bisector is equidistant from A and B.

C

T

A

B

3PerpBisectors Therefore, the circumcenter is equidistant from

each vertex.

Why ?

C

T

A

B

3PerpBisectors

Since C is on each perpendicular bisector it is equidistant from each segment’s endpoint.

Therefore…

C

T

A

B

3PerpBisectorsThe equal distances are radii for a circle that is …

written around (circumscribed) the circle.

Ergo the term… Circumcenter.

C

T

A

B

3PerpBisectorsWhy is this important?

If A, B, and T are three cities, Point C is the ideal place to build a communication tower to broadcast to each city.

TrianglePoints of Concurrency

#2Angle bisectors

and the Incenter.

Angle Bisector

A

B C

Angle Bisector

A

B C

From any general point on the angle bisector, the perpendicular distance to either side is the same.

Why ?P

XY

Angle Bisector

A

B C

The top angles are congruent by definition of angle bisector.

There are two right angles are congruent by def. of perpendicular.

P

AP AP by the reflexive property of =

?? PAX PAY By AAS

XY

PX PY By CPCTC

Angle Bisector

A

B C

Each new point creates 2 congruent triangles by AAS.

Why is this important ?

Let’s see.

AngleBisectors

A

B C

P

Since the incenter P is on each angle bisector….

AngleBisectors

A

B C

P

Point P is equidistant from each side.

Remember, the distance from

a point to a line is the brown perpendicular segment.

Why is this important?

Because these equal distances

are radial distances.

AngleBisectors

A

B C

P

The incenter generates an inscribed circle.

Radial distances refer to a circle.

TrianglePoints of Concurrency

#3Medians

and the Centroid.

A

BC

A median is a segment connecting the vertex to the midpoint of a side of a triangle.

A

BC

P

The 3 medians meet at the centroid – point P.

EF

G

D

3 Medians Centroid

A

B C

Each little triangle is unique, yet they all have something in common.

What is it ?

Area AEF = 7.08 cm2

Area ECF = 7.08 cm2

Area EGC = 7.08 cm2

Area EBG = 7.08 cm2

Area DBE = 7.08 cm2

Area AED = 7.08 cm2

EF

G

D

3 Medians Centroid

E

F

G

D

A

B C

Since the areas of the little triangles around the centroid E are the same, …

the triangle will balance on the centroid.

Mobiles balance objects in an artistic form.

Calder’s Mobile at the East Wing of the National Gallery of ArtIn Washington DC.

Triangles balanced at their centroids.

The birds are tied to their centers of balance or centroids.

TrianglePoints of Concurrency

#4Altitudes

and the Orthocenter.

B

A

C

Acute Triangle

The point of concurrency is called…

The Orthocenter

B

A

C

Right Triangle

The point of concurrency is called…

The Orthocenter

B

A

C

Obtuse Triangle

Notice that although the altitudes are not concurrent…

B

A

C

Obtuse Triangle

The lines containing the altitudes are concurrent.

The Orthocenter

So what do you think is the importance or practical application of the orthocenter is?

Nothing !!! It is just an interesting fact that mathematicians have discovered.

Isn’t this FUN !!! Psych

Summary

bisectors

Line Type ConcurrencyPoint

Importance

bisectors

Medians

Altitudes

Circumcenter

Incenter

Centroid

Orthocenter

Equidistant from the vertices

Circumscribed Circle

Equidistant from the sides

Inscribed Circle

Center of Balance

None

C’est fini.Good day and good luck.

A Senior Citizen Production

That’s all folks.