Geometry Chapter 5

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Geometry Chapter 5

• By Benjamin Koch and
• Satya Nayagam

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Section 5-1 Perpendicular and Angle Bisectors

• Definitions
• Equidistant- a point that is the same distance
  from two or more objects
• Locus- set of points that satisfies a given
  condition

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Section 5-1 Theorems

• 5-1-1 Perpendicular Bisector Theorem- If a point
  is on the perpendicular bisector of a segment,
  then it is equidistant from the endpoints of the
  segment
• 5-1-2 Converse of Perpendicular Bisector
  Theorem- If a point is equidistant from the
  endpoints of a segment, then it is on the
  perpendicular bisector of the segment
• 5-1-3 Angle Bisector Theorem- If a point is on
  the bisector of an angle, then it is equidistant
  from the sides of the angle
• 5-1-4 Converse of Angle Bisector Theorem- If a
  point in the interior of an angle is equidistant
  from the sides of the angle, then it is on the
  bisector of the angle

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Section 5-1 Example
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Section 5-2 Bisectors of Triangle

• Definitions
• Concurrent- 3 or more lines that intersect at one
  point
• Point of Concurrency- point where concurrent
  lines intersect
• Circumcenter- point of intersection of
  perpendicular bisectors of triangle, is
  equidistant from vertices of triangle

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• Circumscribed- circle that contains all vertices
  of triangle
• Incenter- the point of concurrency of the angle
  bisectors
• Inscribed- circle that intersects each line at
  exactly on point

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Section 5-2 Theorems

• 5-2-1 Circumcenter Theorem- The circumcenter of
  a triangle is equidistant from the vertices of
  the triangle
• 5-2-2 Incenter Theorem- The incenter of a
  triangle is equidistant from the sides of the
  triangle

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Section 5-2 Example
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Section 5-3 Medians and Altitudes of Triangles

• Definitions
• Median of Triangle- segment whose endpoints are a
  vertex of triangle and the midpoint of opposite
  side
• Centroid- always inside triangle, point where
  triangular region will balance, intersection of
  medians
• Altitude of Triangle- perpendicular segment from
  vertex to line containing opposite side, triangle
  has 3, can be inside, outside, or on triangle
• Orthocenter of a Triangle- the point of
  congruency of the altitudes

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Section 5-3 Theorem

• 5-3-1 Centroid Theorem- centroid is located 2/3
  of the distance from each vertex to midpoint of
  opposite side

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Section 5-3 Example
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Section 5-4 The Triangle Midsegment Theorem

• Definitions
• Midsegment of a Triangle- a segment that joins
  the midpoints of two sides of the triangle

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Section 5-4 Theorems

• 5-4-1 Triangle Midsegment Theorem- midsegment of
  triangle is parallel to a side of triangle, and
  its length is half the length of that side

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Section 5-4 Example
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Section 5-5 Indirect Proof and Inequalities in
One Triangle

• Definitions
• Indirect Proofs- begin by assuming conclusion is
  false, then show assumption leads to
  contradiction, called proof by contradiction

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Section 5-5 Theorems

• 5-5-1 Angle-Side Relationships in Triangles- If
  two sides of a triangle are not congruent, then
  the larger angle is opposite the longer side
• 5-5-2 Angle-Side Relationships in Triangles- If
  two angles of a triangle are not congruent, then
  the longer side is opposite the larger angle
• 5-5-3 Triangle Inequality Theorem- The sum of
  any two side lengths of a triangle is greater
  than the third side length

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Section 5-5 Example
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Section 5-6 Inequalities in Two Triangles
Theorems

• 5-6-1 Hinge Theorem- if 2 sides of 1 triangle are
  congruent to 2 sides of another triangle and
  included angles arent congruent, then longer 3rd
  side is across from larger included angle,
  Converse larger included angle is across from
  longer 3rd side.

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Section 5-6 Example
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Section 5-7 Pythagorean Theorem Definitions

• Definitions
• Pythagorean Triple- three nonzero whole numbers
  that works out into the three sides of a right
  triangle

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Section 5-7 Pythagorean Theorem Theorems

• 5-7-1 Converse of Pythagorean Theorem-if sum of
  squares of lengths of 2 sides of triangle is
  equal to square of length of 3rd side, then
  triangle is right
• 5-7-2 Pythagorean Inequalities Theorem-if c2 gt
  a2b2 then triangle abc is obtuse, if c2 lt
  a2b2 then triangle abc is acute

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Section 5-7 Example
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Section 5-8 Applying Special Right Triangles
Theorems

• 5-8-1 45-45-90 degrees Triangle Theorem- In a
  45-45-90 triangle, both legs are congruent, and
  the length of the hypotenuse is the length of a
  leg times the square root of 2
• 5-8-2 30-60-90 degrees Triangle Theorem- In a
  30-60-90 triangle, the length of the hypotenuse
  is 2 times the length of the shorter leg, and the
  length of the longer leg is the length of the
  shorter leg times the square root of 3

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Section 5-8 Example