5.3 Concurrent Lines, Medians and Altitudes To Identify Properties of Perpendicular Bisectors and...
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Transcript of 5.3 Concurrent Lines, Medians and Altitudes To Identify Properties of Perpendicular Bisectors and...
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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.
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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.
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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.
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Circumcenter
• The Circumcenter is Equidistant to each vertex of the Triangle
• RC = QC = SC
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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.
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Why use this?
• What is the purpose of a Circumcenter?
• What would this ever be used for?
• Lets look at an example…
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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
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Map of Coasters
• Where would the bathrooms go?
• What shape do the coasters make?
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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
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Example 2
(-4 ,3) is the circumcenter
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Example 3
a) DG b) EK
c) FJ d) DE
19 17
15 19
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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)
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5.3 Concurrent LinesIncenter
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The Incenter
• The Incenter is the point of concurrency of the Angle Bisectors of the Triangle.
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The Incenter
• The Incenter is equidistant to each side of the triangle.
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The Incenter
• The Incenter is the center of a circle you can inscribe inside the triangle.
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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.
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Lets look at the Map!
• Where would be the best location to put the Statue that it would be equidistant to each road?
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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.
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Example 1
a)
b)
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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.
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The CentroidPoint of Concurrency of the Medians of a Triangle
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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.
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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.
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What is so great about the Centroid
• The Centroid is two-thirds the distance from each vertex to the midpoint of the opposite side.
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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??
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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.
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The Orthocenter Point of Concurrency of the Altitudes of a Triangle
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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.
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Median or Altitude?
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So the Orthocenter …
• Is the Point of Concurrency of the Altitudes.
• Can lie inside, outside, or on the triangle.
• Is fun to say.