5 3 Concurrent Lines Medians and Altitudes To

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5. 3 Concurrent Lines, Medians and Altitudes • To Identify Properties of Perpendicular Bisectors

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 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

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

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

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

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

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

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

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 2 (-4 , 3) is the circumcenter

Example 3 a) DG 19 c) FJ 15 b) EK 17 d) DE 19

Example 3 a) DG 19 c) FJ 15 b) EK 17 d) DE 19

So the Circumcenter … • Is the Point of Concurrency of the Perpendicular Bisectors.

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 Lines Incenter

5. 3 Concurrent Lines Incenter

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

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 equidistant to each side of the triangle.

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

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

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

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,

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)

Example 1 a) b)

So the Incenter … • Is the Point of Concurrency of the Angle Bisectors.

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 Centroid Point of Concurrency of the Medians of a Triangle

The Centroid Point of Concurrency of the Medians of a Triangle

What is a Median of a Triangle • The Median of a Triangle is

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 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

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

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. •

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

The Orthocenter Point of Concurrency of the Altitudes of a Triangle

What is an Altitude • An Altitude of a triangle is the perpendicular segment

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?

Median or Altitude?

So the Orthocenter … • Is the Point of Concurrency of the Altitudes. •

So the Orthocenter … • Is the Point of Concurrency of the Altitudes. • Can lie inside, outside, or on the triangle. • Is fun to say.