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2.10 – Bisectors and Midsegments

Objectives

  • Identify and explore the angle bisectors, perpendicular bisectors, and midsegments of triangles.
  • Discover the circumcenters, incenters, and side lengths of triangles and their relationships with angle bisectors, perpendicular bisectors, and midsegments.
  • Prove theorems related to the bisectors and segments of triangles.

 

Key Terms

  • Circumcenter of the Triangle – The point where the perpendicular bisectors of the three sides of a triangle intersect
    • The center of the only circle that can be circumscribed about a given triangle.
    • J is the circumcenter of the diagram below

GeoA 2.10 - Circumcenter Ex1

  • Incenter – The point where the three angle bisectors of a triangle intersect
    • The center of the circle that can be inscribed inside a given triangle.
    • Q is the incenter of the diagram below

GeoA 2.10 - Incenter Ex1

  • Midsegment – A segment connecting two sides of a triangle at their midpoints.
    • \overline{LM} is the midsegment of the triangle below

GeoA 2.10 - Midsegment Ex1

 

Notes

Review
GeoA 2.10 - Bisectors

GeoA 2.10 - Review

Bisectors and Midsegments
GeoA 2.10 - Bisectors Midsegments
Angle Bisectors
  • Bisects a triangle’s angle

GeoA 2.10 - Angle Bisecor

Perpendicular Bisectors
  • Bisects a triangle’s side at 90 degrees

GeoA 2.10 - Perpendicular Bisector

Midsegments
  • The midsegment of \Delta ABC is \overline{LM}.
  • What is the length of \overline{AC} if \overline{LM} is 15 inches long?

GeoA 2.10 - Midsegment Ex2

Answer: 30 inches

Incenters and Circumcenters
  • Incenter
    • The point shared by a triangle’s three angle bisectors
    • The point where a triangle’s three angle bisectors intersect
    • P is the incenter of the triangle below

GeoA 2.10 - Incenter2

  • Circumcenter
    • If a triangle is obtuse, then its circumcenter and its orthocenter both lie outside the triangle.
    • If a triangle is a right triangle, then its circumcenter and its orthocenter both lie on the triangle.
    • If a triangle is acute, then its circumcenter and its orthocenter both lie inside the triangle.

GeoA 2.10 - Circumcenters

Triangle Proportionality & Midsegment Theorems
GeoA - 02.09 Theorems


  • The triangle proportionality theorem works with embedded similar triangles
    • A line parallel to one side of a triangle divides the other two sides proportionally.
    • AA Similarity Theorem
    • The Triangle Proportionality Theorem is NOT a perpendicular line to one side of a triangle.
    • Converse of the Triangle Proportionality Theorem is a perpendicular line to one side of a triangle.

GeoA 2.10 - Triangle Proportionality Theorem


  • Example
    • In this example, \overline{PQ} is parallel to \overline{RS}.
    • The length of \overline{RP} is 4 cm; the length of \overline{PT} is 16 cm; the length of \overline{QT} is 20 cm.
    • What is the length of \overline{SQ}?

GeoA 2.10 - Triangle Proportionality

Answer: 5 cm

Sierpinski Triangle
  • This is an example of the triangle proportionality theorem

GeoA 2.10 - Triangle Midsegment Theorem GeoA 2.10 - Sierpinski

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