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
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,
is the perpendicular bisector.
In the isosceles ∆POQ,
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
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
5-1 Bisectors, Medians, and Altitudes
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
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
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
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.
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
23
23
23L
5-1 Bisectors, Medians, and Altitudes
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
5-1 Bisectors, Medians, and Altitudes
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
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.
The altitudes of a triangle also intersect at a common point called the orthocenter.
Altitudes of an acute triangle.
Altitudes of an obtuse triangle.