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

## Notes

Review

Bisectors and Midsegments
Angle Bisectors
• Bisects a triangle’s angle

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

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

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

• 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.

Triangle Proportionality & Midsegment 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.

• 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 $\overline{SQ}$?