5.3 Concurrent Lines, Medians and Altitudes To Identify Properties of Perpendicular Bisectors and...

5.3 Concurrent Lines, Medians and Altitudes

• To Identify Properties of Perpendicular Bisectors and Angle Bisectors

• To Identify Properties of Medians and Altitudes of a Triangle.

Concurrent Lines

• When three or more lines intersect in one point they are Concurrent.

• The point at which they intersect at is called the Point of Concurrency.

Perpendicular Bisectors

• The Perpendicular Bisectors of a Triangle meet at a point called the Circumcenter

• The Perpendicular Bisectors of the Sides meet at the Circumcenter C.

Circumcenter

• The Circumcenter is Equidistant to each vertex of the Triangle

• RC = QC = SC

Circle it!

• The Circumcenter is also the center of a circle you can draw around or Circumscribe About the Triangle.

• The Distances to the Vertices are the radii of the circle.

Why use this?

• What is the purpose of a Circumcenter?

• What would this ever be used for?

• Lets look at an example…

Where is the Bathroom?

• Great Adventure is building a whole new section to its park with 3 new Roller Coasters.

• The Coaster locations are already set but a Restroom needs to be built so each ride had quick access to it.

• Your job is to find the best possible location of the Restroom

Map of Coasters

• Where would the bathrooms go?

• What shape do the coasters make?

Find the Circumcenter!

• Remember the Circumcenter is the point of concurrency of the Perpendicular Bisectors.

• The Cicumcenter is Equidistant to Every Vertex of the triangle.

• The bathroom would be put at the Circumcenter

Example 2

(-4 ,3) is the circumcenter

Example 3

a) DG b) EK

c) FJ d) DE

19 17

15 19

So the Circumcenter …

• Is the Point of Concurrency of the Perpendicular Bisectors.

• Is Equidistant to each Vertex (Angle) of Triangle.

• Is The Center of a Circle you can Circumscribe about the Triangle.

• Lies either inside (Acute Triangle), Outside (Obtuse Triangle), or on the Hypotenuse (Right Triangle)

5.3 Concurrent LinesIncenter

The Incenter

• The Incenter is the point of concurrency of the Angle Bisectors of the Triangle.

The Incenter

• The Incenter is equidistant to each side of the triangle.

The Incenter

• The Incenter is the center of a circle you can inscribe inside the triangle.

Build a Statue!

• You are to build a statue honoring the Greatest Lyndhurst Swim Coach of all time, Mr. Frew.

• You are to build the statue in a park that is surrounded by three roads. The Mayor wants the statue equidistant to the three roads so all can see.

• Your job is to find the best possible location of the Statue.

Lets look at the Map!

• Where would be the best location to put the Statue that it would be equidistant to each road?

Find the Incenter• By locating the point of concurrency

of the angle bisectors, the Incenter, we find the location that is equidistant to the sides of the triangle.

• The Incenter would be the best Location for the statue of Mr. Frew.

Example 1

a)

b)

So the Incenter …

• Is the Point of Concurrency of the Angle Bisectors.

• Is Equidistant to each Segment (side) of the Triangle.

• Is the Center of a Circle you can Inscribe inside the Triangle.

• Always lies inside the triangle.

The CentroidPoint of Concurrency of the Medians of a Triangle

What is a Median of a Triangle

• The Median of a Triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side.

The Centroid

• The Point of Concurrency of the Medians is called the Centroid.

• The point is also called the center of gravity of a triangle because it’s the point where a triangular shape will balance.

What is so great about the Centroid

• The Centroid is two-thirds the distance from each vertex to the midpoint of the opposite side.

Try this…

• In the Triangle to the left, D is the centroid and BE = 6. Find:• DE

• BD

• What if BD = 12? Find:• DE

• BE

How does DE relate to BD??

So the Centroid …

• Is the Point of Concurrency of the Medians.

• Is two-thirds the distance from each vertex to the midpoint of the opposite side.

• Is the Point of Balance of the Triangle.

• Is always inside the triangle.

The Orthocenter Point of Concurrency of the Altitudes of a Triangle

What is an Altitude

• An Altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side.

• Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or lie outside the triangle.

Median or Altitude?

So the Orthocenter …

• Is the Point of Concurrency of the Altitudes.

• Can lie inside, outside, or on the triangle.

• Is fun to say.