5-2 Perpendicular and Angle Bisectors

Perpendicular Bisectors A point is equidistant from two objects if it is the same distance from each. A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint. Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Using the Perpendicular Bisector Theorem What is the length of AB?

 What is the length of QR?

Angle Bisectors The distance from a point to a line is the length of the perpendicular segment from the point to the line. Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

Using the Angle Bisector Theorem What is the length of RM?

 What is the length of FB?

5-3 Bisectors in Triangles

Concurrency When three of more lines intersect at one point, they are concurrent. The point at which they intersect is the point of concurrency. Concurrency of Perpendicular Bisectors Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Circumcenters The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter. – You can use the circumcenter to circumscribe the triangle. The circumcenter of a triangle can be INSIDE, ON, or OUTSIDE the triangle.

Incenters Concurrency of Angle Bisectors Theorem: The bisectors of the angles of a triangles are concurrent at a point equidistant from the sides of the triangle. The point of concurrency of the angle bisectors of a triangle is called the incenter. – The incenter is the center of the circle that can be inscribed in the triangle.

Using the Incenter of a Triangle GE = 2x – 7 and GF = x + 4. What is GD?

 QN = 5x + 36 and QM = 2x What is QO?