5-1 Bisectors, Medians, and Altitudes 1.) Identify and use perpendicular bisectors and angle bisectors in triangles. 2.) Identify and use medians and altitudes

5-1 Bisectors, Medians, and Altitudes

1.) Identify and use perpendicular bisectors and angle bisectors in triangles.

2.) Identify and use medians and altitudes in triangles.


perpendicular bisector - a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side.

Theorem 5.1 Perpendicular Bisector Theorem

Theorem 5.2 Converse of Perpendicular Bisector Theorem


The perpendicular bisector does not have to start from a vertex!

Example:

In the scalene ∆CDE,

is the perpendicular bisector.

AB

In the right ∆MLN,


In the isosceles ∆POQ,


Since a triangle has three sides, how many perpendicular bisectors do triangles have??

Perpendicular bisectors of a triangle intersect at a common point.


Perpendicular bisectors of a triangle intersect at a common point.

Concurrent Lines:Three or more lines that intersect at a common point.

Point of Concurrency:Point of intersection for concurrent lines.

Circumcenter:Point of concurrency of the perpendicular bisectors

of an triangle.

**The circumcenter does not have to belong inside the triangle.

Circumcenter


Theorem 5.3 Circumcenter Theorem

The circumcenter of a triangle is equidistant from thevertices of the triangle.

**If J is the circumcenter of ABC, then AJ = BJ = CJ.

A

B

C

J

Example 1: BD is the perpendicular bisector of AC. Find AD


Example 2: In the diagram, WX is the perpendicular bisector of YZ.

(a) What segment lengths in the diagram are equal?

(b) Is V on WX?


Example 3: In the diagram, JK is the perpendicular bisector of NL.

(a) Find NK.

(b) Explain why M is on JK.


Example 4: In the diagram, BC is the perpendicular bisector of AD. Find the value of x.



Theorem 5.4 Angle Bisector Theorem

Theorem 5.5 Converse of Angle Bisector Theorem


Theorem 5.6 Incenter Theorem

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

**If G is the incenter of ABC, then GE = GD = GF.

AF is angle bisector of <BAC

BD is angle bisector of <ABC

CE is angle bisector of <BCA

Angle bisectors of a triangle intersect at a common point called the incenter.

Angle bisectors of a triangle are congruent.


Example 5: Find the measure of angle GFJ if FJ bisects <GFH.


Example 6: Find the value of x.


Example 7: Find the value of x.


Example 8: QS is the angle bisector of <PQR.Find the value of x.


Classwork:

Study Guide and Intervention p. 55

Extra problems: p. 242 #6, p. 243 #16


Median -- a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

Since there are three vertices, there are three medians.

In the figure C, E and F are the midpoints of the sides of the triangle.

The medians of a triangle also intersect at a common point called the centroid.


Theorem 5.7 Centroid Theorem

The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

**If L is the centroid of ABC, then AL = AE,

BL = BF

CL = CD

23

23

23L


Example 1: Points U, V, and W are the midpoints of YZ, ZX, and XY. Find a, b, and c.

X

W

Y

U

Z

V

7.4

2a

8.7

15.2

5c

3b + 2


Example 2: Points T, H, and G are the midpoints of MN, MK, and NK. Find w, x, and y.

M

T

K

H

N

G

2x

2.3

4.1

2y

3.2

3w - 2


Altitude -- a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.

Every triangle contains 3 altitudes.

The altitudes of a triangle also intersect at a common point called the orthocenter.

Altitudes of an acute triangle.

Altitudes of an obtuse triangle.

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5-1 Bisectors, Medians, and Altitudes 1.) Identify and use perpendicular bisectors and angle bisectors in triangles. 2.) Identify and use medians and altitudes