Bisectors in Triangles Academic Geometry

Perpendicular Bisectors and Angle Bisectors In the diagram below CD is the perpendicular bisector of AB. CD is perpendicular to AB at its midpoint. AC and CB can be connected to draw triangles. c a d b

Perpendicular Bisectors and Angle Bisectors We learned in Chapter 4 that Triangle CAD is congruent to Triangle CBD. CA congruent CB C is equidistant from points A and B c a d b

Theorem 5-2 If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Theorem 5-3 If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Real World Connection Use the map of Washington, D.C. Describe the set of points that are equidistant from the Lincoln Memorial and the Capitol.

Real World Connection A point that is equidistant from the Lincoln Memorial and the Capitol must be on the perpendicular bisector of the segment whose endpoints are the Lincoln Memorial and the Capitol. Therefore, all points on the perpendicular bisector of the segment whose endpoints are the Lincoln Memorial and the Capitol are equidistant from the Lincoln Memorial and the Capitol.

The distance from a point to a line is the length of the perpendicular segment from the point to the line. AD is the bisector of <CBA If you measure the lengths of the perpendicular segments from D to the sides of the angle you will find the lengths are equal so D is equidistant from the sides. a b c d

Theorem 5-4 Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

Theorem 5-5 Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, the point is on the angle bisector

Using the Angle Bisector Theorem Find x, FB, and FD