- Applythe Concurrencyof AltitudesTheoremto identifythe pointof concurrencyof the altitudesof the
triangle(the orthocenter). - Use the Concurrencyof AltitudesTheoremto solveproblemsinvolvingthe orthocenterof triangles.
Introduction
In this lessonwe will concludeour discussionsaboutspecialline segmentsassociatedwith trianglesby
examiningaltitudesof triangles.We will learnhow to find the locationof a pointwithinthe trianglethat involves
the altitudes.
Definitionof Altitudeof a Triangle
Analtitudeof a triangleis the line segmentfroma vertexperpendicularto the oppositeside.Hereis an
examplethat showsthe altitudefrom vertex in an acutetriangle.
We needto be carefulwith altitudesbecausethey do not alwayslie insidethe triangle.For example,if the
triangleis obtuse,then we can easilysee how an altitudewouldlie outsideof the triangle.Supposethat we
wishedto constructthe altitudefrom vertex in the followingobtusetriangle:
In orderto do this, we mustextendside as follows:
Will the remainingaltitudesfor (thosefrom vertices and ) lie insideor outsideof the triangle?
Answer:The altitudefrom vertex will lie insidethe triangle;the altitudefrom vertex will lie outsidethe
triangle.