Presentation on theme: "5.3 Concurrent Lines, Medians and Altitudes"— Presentation transcript:
15.3 Concurrent Lines, Medians and Altitudes To Identify Properties of Perpendicular Bisectors and Angle BisectorsTo Identify Properties of Medians and Altitudes of a Triangle.
2Concurrent LinesWhen three or more lines intersect in one point they are Concurrent.The point at which they intersect at is called the Point of Concurrency.
3Perpendicular Bisectors The Perpendicular Bisectors of a Triangle meet at a point called the CircumcenterThe Perpendicular Bisectors of the Sides meet at the Circumcenter C.
4CircumcenterThe Circumcenter is Equidistant to each vertex of the TriangleRC = QC = SC
5Circle 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.
6Why use this? What is the purpose of a Circumcenter? What would this ever be used for?Lets look at an example…
7Where 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
8Map of Coasters Where would the bathrooms go? What shape do the coasters make?
9Find 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
12So 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)
14The IncenterThe Incenter is the point of concurrency of the Angle Bisectors of the Triangle.
15The IncenterThe Incenter is equidistant to each side of the triangle.
16The IncenterThe Incenter is the center of a circle you can inscribe inside the triangle.
17Build 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.
18Lets look at the Map!Where would be the best location to put the Statue that it would be equidistant to each road?
19Find the IncenterBy 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.
21So 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.
22Point of Concurrency of the Medians of a Triangle The CentroidPoint of Concurrency of the Medians of a Triangle
23What 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.
24The CentroidThe 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.
25What is so great about the Centroid The Centroid is two-thirds the distance from each vertex to the midpoint of the opposite side.
26Try this…In the Triangle to the left, D is the centroid and BE = 6. Find:DEBDWhat if BD = 12? Find:DEBEHow does DE relate to BD??
27So 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.
28Point of Concurrency of the Altitudes of a Triangle The OrthocenterPoint of Concurrency of the Altitudes of a Triangle
29What is an AltitudeAn 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.