1. 5-2 Bisectors of Triangles
Objectives
Prove and apply properties of
perpendicular bisectors of a triangle.
Prove and apply properties of angle
bisectors of a triangle.
Holt McDougal Geometry
2. 5-2 Bisectors of Triangles
Vocabulary
concurrent
point of concurrency
circumcenter of a triangle
circumscribed
incenter of a triangle
inscribed
Holt McDougal Geometry
3. 5-2 Bisectors of Triangles
Helpful Hint
The perpendicular bisector of a side of a triangle
does not always pass through the opposite
vertex.
Holt McDougal Geometry
4. 5-2 Bisectors of Triangles
When three or more lines intersect at one point, the
lines are said to be concurrent. The point of
concurrency is the point where they intersect. In the
construction, you saw that the three perpendicular
bisectors of a triangle are concurrent. This point of
concurrency is the circumcenter of the triangle.
Holt McDougal Geometry
5. 5-2 Bisectors of Triangles
The circumcenter can be inside the triangle, outside
the triangle, or on the triangle.
Holt McDougal Geometry
6. 5-2 Bisectors of Triangles
The circumcenter of ΔABC is the center of its
circumscribed circle. A circle that contains all the
vertices of a polygon is circumscribed about the
polygon.
Holt McDougal Geometry
7. 5-2 Bisectors of Triangles
Example 1: Using Properties of Perpendicular
Bisectors
DG, EG, and FG are the
perpendicular bisectors of
∆ABC. Find GC.
G is the circumcenter of ∆ABC. By
the Circumcenter Theorem, G is
equidistant from the vertices of
∆ABC.
GC = CB Circumcenter Thm.
GC = 13.4 Substitute 13.4 for GB.
Holt McDougal Geometry
8. 5-2 Bisectors of Triangles
A triangle has three angles, so it has three angle
bisectors. The angle bisectors of a triangle are
also concurrent. This point of concurrency is the
incenter of the triangle .
Holt McDougal Geometry
9. 5-2 Bisectors of Triangles
Remember!
The distance between a point and a
line is the length of the perpendicular
segment from the point to the line.
Holt McDougal Geometry
10. 5-2 Bisectors of Triangles
Unlike the circumcenter, the incenter is always inside
the triangle.
Holt McDougal Geometry
11. 5-2 Bisectors of Triangles
The incenter is the center of the triangle’s inscribed
circle. A circle inscribed in a polygon intersects each
line that contains a side of the polygon at exactly
one point.
Holt McDougal Geometry
12. 5-2 Bisectors of Triangles
Example 3A: Using Properties of Angle Bisectors
MP and LP are angle bisectors of ∆LMN. Find the
distance from P to MN.
P is the incenter of ∆LMN. By the Incenter Theorem,
P is equidistant from the sides of ∆LMN.
The distance from P to LM is 5. So the distance
from P to MN is also 5.
Holt McDougal Geometry
13. 5-2 Bisectors of Triangles
Example 3B: Using Properties of Angle Bisectors
MP and LP are angle bisectors
of ∆LMN. Find m∠PMN.
m∠MLN = 2m∠PLN PL is the bisector of ∠MLN.
m∠MLN = 2(50°) = 100° Substitute 50° for m∠PLN.
m∠MLN + m∠LNM + m∠LMN = 180° Δ Sum Thm.
100 + 20 + m∠LMN = 180 Substitute the given values.
m∠LMN = 60° Subtract 120° from both
sides.
PM is the bisector of ∠LMN.
Substitute 60° for m∠LMN.
Holt McDougal Geometry
15. 5-2 Bisectors of Triangles
Objectives
Apply properties of medians of a
triangle.
Apply properties of altitudes of a
triangle.
Holt McDougal Geometry
16. 5-2 Bisectors of Triangles
Vocabulary
median of a triangle
centroid of a triangle
altitude of a triangle
orthocenter of a triangle
Holt McDougal Geometry
17. 5-2 Bisectors of Triangles
A median of a triangle is a segment whose
endpoints are a vertex of the triangle and the
midpoint of the opposite side.
Every triangle has three medians, and the medians
are concurrent.
Holt McDougal Geometry
18. 5-2 Bisectors of Triangles
The point of concurrency of the medians of a triangle
is the centroid of the triangle . The centroid is
always inside the triangle. The centroid is also called
the center of gravity because it is the point where a
triangular region will balance.
Holt McDougal Geometry
19. 5-2 Bisectors of Triangles
An altitude of a triangle is a perpendicular segment
from a vertex to the line containing the opposite side.
Every triangle has three altitudes. An altitude can be
inside, outside, or on the triangle.
Holt McDougal Geometry
20. 5-2 Bisectors of Triangles
In ΔQRS, altitude QY is inside the triangle, but RX and
SZ are not. Notice that the lines containing the
altitudes are concurrent at P. This point of
concurrency is the orthocenter of the triangle.
Holt McDougal Geometry
21. 5-2 Bisectors of Triangles
Helpful Hint
The height of a triangle is the length of an
altitude.
Holt McDougal Geometry
22. 5-2 Bisectors of Triangles
vocabulary
The midsegment of a
triangle - Segment that
joins the midpoints of
any two sides of a
triangle.
Holt McDougal Geometry
23. 5-2 Bisectors of Triangles
Theorem
The midsegment of a
triangle is half the
length of, and
parallel to, the third
side of a triangle.
Holt McDougal Geometry
