5-1 Perpendicular and Angle Bisectors
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find each measure of MN.
Justify
MN = 2.6
Perpendicular Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Write an equation to solve for a.
Justify
3a + 20 = 2a + 26
Converse of  Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find the measures of BD and
BC. Justify
BD = 12
BC =24
Converse of  Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find the measure of BC.
Justify
BC = 7.2
 Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Write the equation to solve for x. Justify
your equation.
3x + 9 = 7x – 17
 Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Find the measure.
mEFH, given that mEFG = 50°.
Justify
m EFH = 25
Converse of the  Bisector Theorem
Holt Geometry
5-1 Perpendicular and Angle Bisectors
Write an equation in point-slope form for the
perpendicular bisector of the segment with
endpoints C(6, –5) and D(10, 1).
Holt Geometry
Perpendicular Bisectors
of a triangle…
C
• bisect each side at a right angle
• meet at a point called the circumcenter
• The circumcenter is equidistant from the 3 vertices
of the triangle.
• The circumcenter is the center of the circle that is
circumscribed about the triangle.
• The circumcenter could be located inside, outside,
or ON the triangle.
Angle Bisectors
of a triangle…
I
• bisect each angle
• meet at the incenter
• The incenter is equidistant from the 3
sides of the triangle.
• The incenter is the center of the circle that
is inscribed in the triangle.
• The incenter is always inside the circle.
DG, EG, and FG are the perpendicular
bisectors of ∆ABC. Find GC.
GC = 13.4
MZ and LZ are perpendicular bisectors of ∆GHJ. Find GM
GM = 14.5
Z is the circumcenter of ∆GHJ. GK and JZ
GK = 18.6
JZ = 19.9
Find the circumcenter of ∆HJK with vertices
H(0, 0), J(10, 0), and K(0, 6).
MP and LP are angle bisectors of ∆LMN. Find the
distance from P to MN.
MP and LP are angle bisectors
of ∆LMN. Find mPMN.
mPMN = 30
5-3: Medians
and Altitudes
and Angle
Bisectors B
5-1 Perpendicular
Medians
of triangles:
X
P
Y
•Endpoints are a vertex
A
C
Z
and midpoint of opposite side.
•Intersect at a point called the centroid
•Its coordinates are the average of the 3
vertices.
•The centroid is ⅔ of the distance from
each vertex to the midpoint of the opposite
2
2
2
AP

AY
BP

BZ
CP

CX
side.
3
3
3
•The centroid is always located inside the
triangle.
Holt Geometry
5-3: Medians
and Altitudes
and Angle
Bisectors
5-1 Perpendicular
Altitudes of a triangle:
• A perpendicular segment from a vertex to
the line containing the opposite side.
• Intersect at a point called the
orthocenter.
• An altitude can be inside, outside, or on
the triangle.
Holt Geometry
In ∆LMN, RL = 21 and SQ =4.
Find LS.
LS = 14
In ∆LMN, RL = 21 and SQ =4.
Find NQ.
12 = NQ
In ∆JKL, ZW = 7, and LX = 8.1.
Find KW.
KW = 21
Example 2: Problem-Solving Application
A sculptor is shaping a
triangular piece of iron that
will balance on the point of a
cone. At what coordinates will
the triangular region balance?
Find the average of the x-coordinates and the
average of the y-coordinates of the vertices of
∆PQR. Make a conjecture about the centroid of a
triangle.
Find the orthocenter of ∆XYZ with vertices
X(3, –2), Y(3, 6), and Z(7, 1).
X