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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
is circumscribed about the triangle.
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
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