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Geometry
Lesson Notes 5.1
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Objective: Identify and use perpendicular bisectors, angle bisectors, and medians in triangles.
Perpendicular bisector: a line, segment, or ray that passes through the midpoint of the side
of a triangle and is perpendicular to that side.
Every triangle has three perpendicular bisectors.
(Points on Perpendicular Bisectors)
(Theorem 5.1: Any point on the perpendicular bisector of a segment is equidistant from the
endpoints of the segment.)
This theorem is the basis of the construction of a perpendicular bisector of a segment.
(Theorem 5.2: Any point equidistant from the endpoints of a segment lies on the
perpendicular bisector of the segment.)
(Perpendicular bisector (alternative definition using theorem): the locus (set) of all points in a
plane equidistant from the endpoints of a given segment.)
Concurrent lines: three or more lines that intersect at a common point. Their point of
intersection is called the point of concurrency.
The three perpendicular bisectors of a triangle are concurrent. Their point of concurrency is
called the circumcenter of the triangle.
Theorem 5.3 Circumcenter Theorem: The circumcenter of a triangle is equidistant from the
vertices of the triangle.
The circumcenter can be used to circumscribe a circle about a triangle.
 The circumcenter of a triangle can lie outside of the triangle.
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Angle bisector: a line, segment, or ray that divides an angle into two congruent angles.
Every triangle has three angle bisectors.
(Points on Angle Bisectors)
(Theorem 5.4: Any point on the angle bisector is equidistant from the sides of
the angle.)
(This theorem is the basis of the construction of an angle bisector of an angle.)
(Theorem 5.5: Any point equidistant from the sides of an angle lies on the angle bisector.)
The three angle bisectors of a triangle are concurrent. Their point of concurrency is called the
incenter of the triangle.
Theorem 5.3 Incenter Theorem: The incenter of a triangle is equidistant from each side of
the triangle.
The incenter of a triangle can be used to inscribe a circle in the triangle.
Median: a segment with one endpoint at the vertex of a triangle and its other endpoint at the
midpoint of the side opposite the vertex.
Every triangle has three medians.
The three medians of a triangle are concurrent. Their point of concurrency is called the
centroid of the triangle.
Theorem 5.3 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 the
median.
The centroid of a triangle is the point of balance of the triangle.
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Example 2 (p 240): Segment Measure Using Medians
Points U, V, and W are midpoints of the sides of XYZ. Find a, b, and c.
Y
7.4
W
U
8.7
5c
3b + 2
15.2
2a
X
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V
Z
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Altitude: a segment from a vertex to the line containing the opposite side and perpendicular
to the line containing the side.
Every triangle has three altitudes.
The three altitudes of a triangle are concurrent. Their point of concurrency is called the
orthocenter of the triangle.
The orthocenter of a triangle can lie outside of the triangle.
The orthocenter of a triangle can lie at the vertex of the right triangle.
Activity: Identify possible locations for circumcenter, incenter, centroid, and
orthocenter (inside the triangle, at a vertex, outside the triangle).
Draw four copies of an acute triangle, a right triangle, and an obtuse triangle.
Draw (construct) a set of each type of special segments for each type of
triangle.
Discuss results.
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E
Example 3 (pp 241-242): Use a System of Equations for Find a Point
Systems of equations can be used to find the coordinates of the orthocenter,
circumcenter, and centroid of a triangle graphed on a coordinate plane.
 CW p 242 # 1, 2 (sample above), 3, 5, 6 (good application of properties),
 HW
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A1a pp 243-244 #11, 13-16, 21-26
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