Geometry Lesson 5 1 Bisectors of Triangles Objective

Geometry Lesson 5 – 1 Bisectors of Triangles Objective: Identify and use perpendicular bisectors in triangles. Identify and use angle bisectors in triangles.

Perpendicular Bisector Perpendicular bisector l Any segment, line, or plane that intersects a segment at its midpoint forming a right angle.

Theorems Perpendicular Bisector Theorem l If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of Perpendicular Bisector Theorem l If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Find AB AB = 4. 1

Find WY WY = 3

Find RT RQ = RT 2 x + 3 = 4 x – 7 3 = 2 x – 7 10 = 2 x 5=x RT = 4 x – 7 = 4(5) – 7 = 20 – 7 = 13

Concurrent lines l 3 or more lines intersect at a point Point of Concurrency l The point where 3 or more lines intersect.

Perpendicular Bisectors Right: On the Triangle Acute: Interior Obtuse: Exterior

Circumcenter l The point of concurrency of the perpendicular bisectors Circumcenter Theorem l The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.

Angle Bisector Angle bisector l A line, segment, or ray that cuts an angle into 2 congruent parts.

Theorem Angle Bisector Theorem l If a point is on the bisector of an angle, then it is equidistant from the sides of that angle. Converse Angle Bisector Theorem l If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

Find XY XY = 7

Find the measure of angle JKL 37

Find SP 3 x + 5 = 6 x – 7 5 = 3 x – 7 12 = 3 x 4=x SP = 6 x – 7 = 6(4) – 7 = 17 SP = 17

Angle bisectors of a triangle Notice all angle bisectors go through a vertex and Intersect in the interior of the triangle.

Incenter l Incenter The point of concurrency of the angle bisectors of a triangle. Incenter Theorem l The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from each side of the triangle.

Find each measure if J is the incenter. JF JF = JE How can we find JE? (JE)2 + 122 = 152 (JE)2 + 144 = 225 (JE)2 = 81 JE = 9 JF = 9

If P is the incenter find the following. PK PK = PJ (PJ)2 + 122 = 202 (PJ)2 + 144 = 400 (PJ)2 = 256 PJ = 16 PK = 16 62 31 27 54

Homework Pg. 327 1 – 8 all, 10 – 14 E, 18 – 34 E, 60 – 64 E