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Lesson 5-1Lesson 5-1Bisectors, Medians, Bisectors, Medians,
and Altitudesand Altitudes
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Ohio Content Standards:
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Ohio Content Standards:
• Formally define geometric figures.
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Ohio Content Standards:• Formally define and explain key aspects of
geometric figures, including:a. interior and exterior angles of polygons;b. segments related to triangles (median, altitude, midsegment);c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);
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Perpendicular Bisector
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Perpendicular Bisector
A line, segment, or ray that passes through the midpoint of the side of a triangle and is perpendicular to that side.
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Theorem 5.1
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Theorem 5.1
Any point on the perpendicular bisector of a segment is equidistant from
the endpoints of the segment.
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Example
C D
A
B
. and
then, bisects and If
BDBCADAC
CDABCDAB
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Theorem 5.2
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Theorem 5.2
Any point equidistant from the endpoints of a segment
lies on the perpendicular bisector of the segment.
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Example
C D
A
B
. ofbisector lar perpendicu on the lies then , If
. ofbisector lar perpendicu on the lies then ,C If
CDBBDBC
CDAADA
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Concurrent Lines
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Concurrent Lines
When three or more lines intersect at a common point.
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Point of Concurrency
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Point of Concurrency
The point of intersection where three or more lines
meet.
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Circumcenter
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Circumcenter
The point of concurrency of the perpendicular bisectors
of a triangle.
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Theorem 5.3Circumcenter Theorem
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Theorem 5.3Circumcenter Theorem
The circumcenter of a triangle is equidistant from the vertices of the triangle.
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Example
CA
B
.then
, ofer circumcent theis If
CKBKAK
ABCK
circumcenter
K
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Theorem 5.4
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Theorem 5.4Any point on the angle
bisector is equidistant from the sides of the angle.
A C
B
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Theorem 5.5
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Theorem 5.5Any point equidistant from
the sides of an angle lies on the angle bisector.
A
B
C
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Incenter
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Incenter
The point of concurrency of the angle bisectors.
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Theorem 5.6Incenter Theorem
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Theorem 5.6Incenter Theorem
The incenter of a triangle is equidistant from each side of
the triangle.
CA
Bincenter
KP
Q
R
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Theorem 5.6Incenter Theorem
The incenter of a triangle is equidistant from each side of
the triangle.
CA
Bincenter
KP
Q
R
If K is the incenter of ABC, then KP = KQ
= KR.
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Median
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Median
A segment whose endpoints are a vertex of a triangle and
the midpoint of the side opposite the vertex.
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Centroid
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Centroid
The point of concurrency for the medians of a triangle.
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Theorem 5.7Centroid Theorem
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Theorem 5.7Centroid Theorem
The centroid of a triangle is located two-thirds of the
distance from a vertex to the midpoint of the side opposite
the vertex on a median.
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Example
CA
B
D L E
F
centroid
.3
2 and ,
3
2
,3
2 , of centroid theis If
CDCLBFBL
AEALABCL
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Altitude
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Altitude
A segment from a vertex in a triangle to the line
containing the opposite side and perpendicular to the line
containing that side.
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Orthocenter
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Orthocenter
The intersection point of the altitudes of a triangle.
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Example
CA
B
D
L
E
F
orthocenter
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Points U, V, and W are the midpoints of YZ, ZX, and XY, respectively. Find a, b, and c.
Y
W U
XV Z
7.45c 8.7
15.22a
3b + 2
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The vertices of QRS are Q(4, 6), R(7, 2), and S(1, 2). Find
the coordinates of the orthocenter of QRS.
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Assignment:
Pgs. 243-245 13-20 all