GeometryLesson 5 – 1

Bisectors of Triangles

Objective:Identify and use perpendicular bisectors in triangles.

Identify and use angle bisectors in triangles.

Perpendicular BisectorPerpendicular bisectorAny segment, line, or plane that intersects

a segment at its midpoint forming a right angle.

TheoremsPerpendicular 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 Perpendicular Bisector Theorem 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 2x + 3 = 4x – 7 3 = 2x – 7 10 = 2x 5 = x

RT = 4x – 7 = 4(5) – 7 = 20 – 7 = 13

Concurrent lines3 or more lines intersect at a point

Point of ConcurrencyThe point where 3 or more lines intersect.

Perpendicular Bisectors

Acute: Interior

Right:On theTriangle

Obtuse:Exterior

CircumcenterCircumcenterThe point of concurrency of the

perpendicular bisectors

Circumcenter TheoremThe perpendicular bisectors of a triangle

intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.

Angle BisectorAngle bisectorA line, segment, or ray that cuts an angle

into 2 congruent parts.

TheoremAngle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant

from the sides of that angle.

Converse Angle Bisector Theorem 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

3x + 5 = 6x – 7 5 = 3x – 7 12 = 3x 4 = x

SP = 6x – 7 = 6(4) – 7 = 17

SP = 17

Angle bisectors of a triangle

Notice all angle bisectors go through a vertex andIntersect in the interior of the triangle.

IncenterIncenterThe point of concurrency of the angle

bisectors of a triangle.

Incenter TheoremThe 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 = JEHow can we find JE?

(JE)2 + 122 = 152

(JE)2 + 144 = 225(JE)2 = 81

JE = 9JF = 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

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