# Concurrent Lines, Medians, and Altitiudes - PowerPoint PPT Presentation

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## Concurrent Lines, Medians, and Altitiudes

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### Point of Concurrency the point at which the lines intersect. ... The perpendicular bisectors of the sides ... Obtuse Triangle: Altitude outside. Orthocenter ... – PowerPoint PPT presentation

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Title: Concurrent Lines, Medians, and Altitiudes

1
Concurrent Lines, Medians, and Altitiudes
2
Concurrent Lines
• When three or more lines intersect in one point,
they are concurrent.
• Point of Concurrency the point at which the
lines intersect.
• For a triangle, there are four different sets of
concurrent lines.

3
Circumcenter
• The perpendicular bisectors of the sides of a
triangle are concurrent (all intersect) at a
point equidistant from the vertices.
• The point of concurrency is called the
circumcenter of the triangle.

4
Picture of the Circumcenter
CH CJ CG
C is the circumcenter.
Using a compass with the point at C, a circle can
be drawn that passes thru G, H and J. The circle
C
5
Incenter
• The bisectors of the angles of a triangle are
concurrent (all intersect) at a point equidistant
from the sides.
• The point of concurrency is called the incenter
of the triangle.

6
Picture of the Incenter
IY IR IW
I is the incenter.
Using a compass with the point at I, a circle can
be drawn that passes thru Y, R and W. The circle
is inscribed in the triangle.
I
7
Medians
• Median of a Triangle A median is a segment
drawn from the vertex of a triangle to the
midpoint of the opposite side.

Median
8
Centroid
• The medians of a triangle are concurrent (all
intersect) at a point that divides each median
into two segments, one of which is twice as long
as the other.
• The point of concurrency is called the centroid.
• Also called the center of gravity or center of
balance.

9
Picture of Centroid
2x
8
k
5
b
C
2b
2k
10
x
4
10
Altitudes
• Altitude of a Triangle An altitude is a segment
drawn from the vertex of a triangle perpendicular
to the opposite side.
• Can be inside, on a side, or outside the traingle

11
Orthocenter
• The lines that contain the altitudes of a
triangle are concurrent (they all intersect at a
single point).
• The point of concurrency of the altitudes is
called the orthocenter.

12
Orthocenter
13
Finding Lengths of Medians
• M is the centroid of DWOR, and WM 16. Find WX.

WX 24
14
Identifying Medians and Altitudes
• Is KX a median, an altitude, neither, or both?

Both