Unit 5Unit 5

Bisectors, Medians, and Altitudes

Bisectors, Medians, and Altitudes

Part 1Part 1

VocabVocab

• Concurrent Lines• When three or more lines intersect

they are called concurrent lines

• Point of Concurrency• The point of intersection for

concurrent lines

• Concurrent Lines• When three or more lines intersect

they are called concurrent lines

• Point of Concurrency• The point of intersection for

concurrent lines

Perpendicular BisectorPerpendicular Bisector

• A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. • Any point on a perpendicular bisector is

equidistant from the endpoints of the segment.(theorem 5.1)

• Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment. (theorem 5.2)

• A triangle has three perpendicular bisectors.

• A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. • Any point on a perpendicular bisector is

equidistant from the endpoints of the segment.(theorem 5.1)

• Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment. (theorem 5.2)

• A triangle has three perpendicular bisectors.

Circumcenter Circumcenter

• The intersection of the three perpendicular bisectors, their point of concurrency.

• The circumcenter of a triangle is equidistant from the vertices of a triangle. (Theorem 5.3)

• The intersection of the three perpendicular bisectors, their point of concurrency.

• The circumcenter of a triangle is equidistant from the vertices of a triangle. (Theorem 5.3)

j

k lLines j, k, and l are ⊥ .bisctors

= = This means AG GC GB

:Also = & = & = BE CE AF CF AD BD

: Also all⊥ bisctors form right angles with the segments they. bisect

G

F

E

D

B

A

C

Angle Bisector TheoremsAngle Bisector Theorems

• Any point on the angle bisector is equidistant from the sides of the angle. (theorem 5.4)

• Any Point equidistant from the sides of an angle lies on the angle bisector. (theorem 5.5)

• Any point on the angle bisector is equidistant from the sides of the angle. (theorem 5.4)

• Any Point equidistant from the sides of an angle lies on the angle bisector. (theorem 5.5)

IncenterIncenter

• The intersection of the three angle bisectors.

• The incenter of a triangle is equidistant from each side of the triangle.

• The intersection of the three angle bisectors.

• The incenter of a triangle is equidistant from each side of the triangle.

jk

l

65

43

21

Segments j, k, and l are allangle bisectors.

This means BF = DF = EF

Also: <1 = <2, <3 = <4, <5 = <6

Also: GH = GII

H

DB

E

F

C

A

G

MedianMedian

• A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

• A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

BD is a median of ABCAD ≅ DC

D

B

A C

CentroidCentroid

• The point of intersection of the three medians of a triangle

• The centroid is the point of balance for any triangle

• The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (Theorem 5.7)

• The point of intersection of the three medians of a triangle

• The centroid is the point of balance for any triangle

• The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (Theorem 5.7)

Centroid ExampleCentroid ExampleBD , AF, CE is a median of ABCtherefore:

AG = 2

3AF and GH =

1

3AF

BG = 2

3BD and GD =

1

3BG

CG = 2

3CE and GE =

1

3CE

G

D

FE

B

A C

////

2

3BP =BE

23(10) =BE

203

=BE

2(EN )=AE2(x) =x+17x=17

AN =AE + ENAN =x+17 + xAN =(17) +17 + (17)AN =51

2

3

1

3

AltitudeAltitude

• A segment formed from the vertex to the line containing the opposite side and is perpendicular to the opposite side.

• Every triangle has three altitudes, their intersection is know as the orthocenter.

• A segment formed from the vertex to the line containing the opposite side and is perpendicular to the opposite side.

• Every triangle has three altitudes, their intersection is know as the orthocenter. Segment BD is an

altitude, therefore∠ BDC and∠ BDA are

both right angles

D

B

A

C

Triangle InequalitiesTriangle Inequalities

Part 2Part 2

Exterior Angle Inequality Theorem

Exterior Angle Inequality Theorem

• If an angle is an exterior angle of a triangle then its measure is greater than the measure of either of its corresponding remote interior angles.

(theorem 5.8)

• If an angle is an exterior angle of a triangle then its measure is greater than the measure of either of its corresponding remote interior angles.

(theorem 5.8)

m∠4 > m∠1

m∠4 > m∠243

2

1

Triangle Sides and Angles Triangle Sides and Angles

• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.

• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.

• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

ExampleExample

9

7

5

B

A

C

∠C is the largest angle

∠B is the middle angle

∠A is the smallest angle

38°65°

77°

E

DF

DF is the largest side

DE is the middle side

EF is the smallest side

Triangle Inequality Theorem

Triangle Inequality Theorem

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • Always check using the two smallest

sides, they must be larger than the third. If this is true the numbers will represent a triangle.

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • Always check using the two smallest

sides, they must be larger than the third. If this is true the numbers will represent a triangle.

Example Example

• Do these numbers represent a triangle?1.) 9, 7, 12

Yes2.) 5, 5, 10

No3.) 1, 4, 6

No4.) 6, 6, 2

Yes

• Do these numbers represent a triangle?1.) 9, 7, 12

Yes2.) 5, 5, 10

No3.) 1, 4, 6

No4.) 6, 6, 2

Yes