Transcript Lesson 5-1 Bisectors, Medians, and Altitudes Ohio Content Standards: Ohio Content Standards: • Formally define geometric figures. Ohio Content Standards: • Formally define and explain key aspects.

Lesson 5-1
Bisectors, Medians,
and Altitudes
Ohio Content Standards:
Ohio Content Standards:
• Formally define geometric figures.
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);
Perpendicular Bisector
Perpendicular Bisector
A line, segment, or ray that
passes through the midpoint
of the side of a triangle and
is perpendicular to that side.
Theorem 5.1
Theorem 5.1
Any point on the
perpendicular bisector of a
segment is equidistant from
the endpoints of the
segment.
Example
A
C
D
B
If AB  CD and AB bisects CD, then
AC  AD and BC  BD.
Theorem 5.2
Theorem 5.2
Any point equidistant from
the endpoints of a segment
lies on the perpendicular
bisector of the segment.
Example
A
C
D
B
If AC  AD, then A lies on theperpendicular bisectorof CD.
If BC  BD, then B lies on theperpendicular bisector of CD.
Concurrent Lines
Concurrent Lines
When three or more lines
intersect at a common point.
Point of Concurrency
Point of Concurrency
The point of intersection
where three or more lines
meet.
Circumcenter
Circumcenter
The point of concurrency of
the perpendicular bisectors
of a triangle.
Theorem 5.3
Circumcenter Theorem
Theorem 5.3
Circumcenter Theorem
The circumcenter of a
triangle is equidistant from
the vertices of the triangle.
Example
B
A
circumcenter
K
C
If K is thecircumcenter of ABC,
then AK  BK  CK .
Theorem 5.4
Theorem 5.4
Any point on the angle
bisector is equidistant from
the sides of the angle.
B
A
C
Theorem 5.5
Theorem 5.5
Any point equidistant from
the sides of an angle lies on
the angle bisector.
B
A
C
Incenter
Incenter
The point of concurrency of
the angle bisectors.
Theorem 5.6
Incenter Theorem
Theorem 5.6
Incenter Theorem
The incenter of a triangle is
equidistant from each side of
the triangle.
incenter
B
P
Q
K
A
R
C
Theorem 5.6
Incenter Theorem
The incenter of a triangle is
equidistant from each side of
the triangle.
incenter
B
P
Q
K
A
R
If K is the incenter
of ABC, then KP =
KQ = KR.
C
Median
Median
A segment whose endpoints
are a vertex of a triangle and
the midpoint of the side
opposite the vertex.
Centroid
Centroid
The point of concurrency for
the medians of a triangle.
Theorem 5.7
Centroid Theorem
Theorem 5.7
Centroid 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.
Example
B
D
centroid
E
L
A
F
C
2
If L is t hecent roidof ABC, AL  AE,
3
2
2
BL  BF , and CL  CD.
3
3
Altitude
Altitude
A segment from a vertex in a
triangle to the line containing
the opposite side and
perpendicular to the line
containing that side.
Orthocenter
Orthocenter
The intersection point of the
altitudes of a triangle.
Example
B
E
D
L
A
F
orthocenter
C
Points U, V, and W are the
midpoints of YZ, ZX, and XY,
respectively. Find a, b, and c.
Y
W
7.4
U
2a
X
V
Z
The vertices of QRS are
Q(4, 6), R(7, 2), and S(1, 2). Find
the coordinates of the
orthocenter of QRS.
Assignment:
Pgs. 243-245
13-20 all