5-3 Bisectors in Triangles
Objective
To identify properties of perpendicular bisectors and angle bisectors
Lesson Vocabulary
- concurrent
- point of concurrency
- circumcenter of a triangle
- circumscribed about
- incenter of a triangle
- inscribed in
In the Solve It, the three lines you drew intersect at one point, the center of the circle. When three or more lines intersect at one point, they are concurrent. The point at which they intersect is the point of concurrency.
Essential Understanding For any triangle, certain sets of lines are always concurrent. Two of these sets of lines are the perpendicular bisectors of the triangle's three sides and the bisectors of the triangle's three angles.
Take Note Theorem 5-6: Concurrency of Perpendicular Bisectors Theorem
Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Diagram
Symbols
Perpendicular bisectors
The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.
Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. You say the circle is circumscribed about the triangle.
Table of Contents
- 6-1 The Polygon Angle-Sum Theorems
- 6-2 Properties of Parallelograms
- 6-3 Proving That a Quadrilateral Is a Parallelogram
- 6-4 Properties of Rhombuses, Rectangles, and Squares
- 6-5 Conditions for Rhombuses, Rectangles, and Squares
- 6-6 Trapezoids and Kites
- 6-7 Polygons in the Coordinate Plane
- 6-8 and 6-9 Coordinate Geometry and Coordinate Proofs