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5-1 Bisectors, Medians, and Altitudes

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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 in triangles.

5-1 Bisectors, Medians, and Altitudes

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

5-1 Bisectors, Medians, and Altitudes

B

A

In the isosceles ∆POQ,

is the perpendicular bisector.

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

Example:

In the scalene ∆CDE,

is the perpendicular bisector.

In the right ∆MLN,

is the perpendicular bisector.

Since a triangle has three sides,

how many perpendicular bisectors do triangles have??

Perpendicular bisectors of a triangle intersect

at a common point.

5-1 Bisectors, Medians, and Altitudes

Circumcenter

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.

5-1 Bisectors, Medians, and Altitudes

Theorem 5.3 Circumcenter Theorem

The circumcenter of a triangle is equidistant from the

vertices of the triangle.

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

B

J

C

A

5-1 Bisectors, Medians, and Altitudes

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

5-1 Bisectors, Medians, and Altitudes

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?

5-1 Bisectors, Medians, and Altitudes

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

(a) Find NK.

(b) Explain why

M is on JK.

5-1 Bisectors, Medians, and Altitudes

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

5-1 Bisectors, Medians, and Altitudes

Theorem 5.4 Angle Bisector Theorem

Theorem 5.5 Converse of Angle Bisector Theorem

5-1 Bisectors, Medians, and Altitudes

Angle bisectors of a triangle are congruent.

Angle bisectors of a triangle intersect

at a common point called the incenter.

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

5-1 Bisectors, Medians, and Altitudes

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

5-1 Bisectors, Medians, and Altitudes

Example 6: Find the value of x.

5-1 Bisectors, Medians, and Altitudes

Example 7: Find the value of x.

5-1 Bisectors, Medians, and Altitudes

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

Find the value of x.

5-1 Bisectors, Medians, and Altitudes

Classwork:

Study Guide and Intervention p. 55

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

5-1 Bisectors, Medians, and Altitudes

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.

5-1 Bisectors, Medians, and Altitudes

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

2

3

2

3

2

3

L

5-1 Bisectors, Medians, and Altitudes

Y

7.4

U

W

8.7

5c

3b + 2

15.2

2a

Z

X

V

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

5-1 Bisectors, Medians, and Altitudes

M

2y

T

H

3.2

4.1

2x

3w - 2

2.3

N

K

G

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

5-1 Bisectors, Medians, and Altitudes

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.

Altitudes of an acute triangle.

Altitudes of an obtuse triangle.

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