Relationships in Triangles

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Relationships in Triangles

• Bisectors, Medians, and Altitudes

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Objectives of this lesson

• To identify and use perpendicular
• bisectors angle bisectors in triangles
• To identify and use medians altitudes in
  triangles

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Vocabulary

• Perpendicular Bisectors
• Angle Bisectors
• Medians
• Altitudes
• Points of Concurrency

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Perpendicular bisector

• A line segment or a ray that passes through the
  midpoint of a side of a triangle is ? to that
  side.

In the picture to the right, the red line segment
is the ? bisector
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Perpendicular Bisector (cont)

• For every triangle there are 3 perpendicular
  bisectors
• The 3 perpendicular bisectors intersect in a
  common
• point named the circumcenter.

In the picture to the right point K is the
circumcenter.
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Perpendicular Bisector (cont)

• Any point on the perpendicular bisector of a
  segment is equidistant
• from the endpoints of the segment
• Any point equidistant from the endpoints of a
  segment lies on the
• perpendicular bisector of the segment

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Angle Bisector

• A line, line segment or ray that bisects an
  interior angle of a triangle

In the picture to the right, the red line segment
is the angle bisector. The ? arc marks show the 2
? angles that were formed when the angle bisector
bisected the original angle.
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Angle Bisector (cont)

• For every triangle there are 3 angle bisectors.
• The 3 angle bisectors intersect in a common point
  named the incenter

In the picture to the right, point I is the
incenter.
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Angle Bisector (cont)

• Any point on the angle bisector is equidistant
  from the sides of the angle.
• Any point equidistant from the sides of an angle
  lies on the angle bisector.

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Median

• A line segment whose endpoints are a vertex of a
• triangle and the midpoint of the side opposite
  the
• vertex.

In the picture to the right, the blue line
segment is the median.
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Median (cont)

• For every triangle there are 3 medians
• The 3 medians intersect in a common point named
  the centroid

In the picture to the right, point O is the
centroid.
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Altitudes
A line segment from a vertex to the line
containing the opposite side and perpendicular to
the line containing that side.
In the picture above, ?ABC is an obtuse triangle
?ACB is the obtuse angle. BH is an altitude.
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Altitudes (cont)

• For every triangle there are 3 altitudes
• The 3 altitudes intersect in a common point
  called
• the orthocenter.

In the picture to the right, point H is the
orthocenter.
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Points of Concurrency
Concurrent Lines 3 or more lines that intersect
at a common point
Point of Concurrency The point of intersection
when 3 or more lines intersect.
Type of Line Segments Point of Concurrency Perpend
icular Bisectors Circumcenter Angle
Bisectors Incenter Median Centroid Altitude
Orthocenter
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Points of Concurrency (cont)

• Facts to remember
• The circumcenter of a triangle is equidistant
  from the
• vertices of the triangle.
• Any point on the angle bisector is equidistant
  from the sides of the angle (Converse of 3)
• Any point equidistant from the sides of an angle
  lies on
• the angle bisector. (Converse of 2)
• The incenter of a triangle is equidistant from
  each side of the triangle.
• The distance from a vertex of a triangle to the
  centroid is 2/3 of the medians entire length.
  The length from the centroid to the midpoint is
  1/3 of the length of the median.

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Points of Concurrency (cont)
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Facts To Remember MEMORIZE!

• 1. Perpendicular Bisectors
• 2. Angle Bisectors
• 3. Medians
• 4. Altitudes

• 1. form right angles AND 2 ? lines segments
• 2. form 2 ? angles
• 3. form 2 ? line segments
• 4. form right angles

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The End
(Finally!)