Median and Altitude of a Triangle Sec 5.3 Goal: To use properties of the medians of a triangle. To...
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Transcript of Median and Altitude of a Triangle Sec 5.3 Goal: To use properties of the medians of a triangle. To...
Median and Altitude of a TriangleSec 5.3
Goal: To use properties of the medians of a triangle. To use properties of the altitudes of a triangle.
Median of a Triangle
Median of a Triangle – a segment whose endpoints are the vertex of a triangle and the midpoint of the opposite side.
Median
Vertex
Median of an Obtuse Triangle
A
CB
P
Point of concurrency “P” or centroid
F
ED
Medians of a TriangleTheorem 5.7
A
CB
P
F
ED
The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side.
If P is the centroid of ABC, then 2AP= 3AF
2 2CP= and BP= 3 3CE BD
Example - Medians of a Triangle
A
CB
P
F
ED
. 5
P is the centroid of ABCPFFind AF and AP
5
Median of an Acute Triangle
A
CB
P
Point of concurrency “P” or centroid
F
DE
Median of a Right Triangle
A
CB
P
Point of concurrency “P” or centroid
The three medians of an obtuse, acute, and a right triangle always meet inside the triangle.
D
E F
Altitude of a Triangle
A
CB
altitude
Altitude of a triangle – the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side
Altitude of an Acute Triangle
A
CB
P
Point of concurrency “P” or orthocenter
The point of concurrency called the orthocenter lies inside the triangle.
Altitude of a Right Triangle
A
CB P
Point of concurrency “P” or orthocenter
The point of concurrency called the orthocenter lies on the triangle.
The two legs are the altitudes
Altitude of an Obtuse Triangle
The point of concurrency of the three altitudes is called the orthocenter
A
CB
P
alt
itude
altitude
The point of concurrency lies outside the triangle.
Altitudes of a TriangleTheorem 5.8
A
CB
P
alt
itude
altitude
The lines containing the altitudes of a triangle are concurrent.
.
, , , , ,
If AE BF and CD are the altitudes of ABCthen the lines AE BF and CD intersect at P
2323323323323323323323323323323323323323 32323333333333333 3
F
E
D
altitu
de
Example
Determine if EG is a perpendicular bisector, and angle bisector, a median, or an altitude of triangle DEF given that:
.a DG FG
.b EG DF
.c DEG FEG
.d and DG FGEG DF
E
D G F
.e DEG FGE
Review Properties / Points of Concurrency
Median -- Centroid
Altitude -- Orthocenter
Perpendicular Bisector --
Circumcenter
Angle Bisector -- Incenter