The document discusses various bisectors and segments related to triangles, including: - The perpendicular bisectors and angle bisectors of a triangle are concurrent, intersecting at the circumcenter and incenter respectively. - A triangle has three medians and three altitudes. The medians intersect at the centroid, while the altitudes intersect at the orthocenter. - Key properties are discussed, such as the circumcenter theorem relating the circumcenter to triangle vertices, and the incenter theorem relating the incenter to triangle sides.

1. 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

2. 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

3. 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.
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4. 5-2 Bisectors of Triangles
Vocabulary
concurrent
point of concurrency
circumcenter of a triangle
circumscribed
incenter of a triangle
inscribed
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5. 5-2 Bisectors of Triangles
Helpful Hint
The perpendicular bisector of a side of a triangle
does not always pass through the opposite
vertex.
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6. 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.
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7. 5-2 Bisectors of Triangles
The circumcenter can be inside the triangle, outside
the triangle, or on the triangle.
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8. 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.
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9. 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
GC = 13.4
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Circumcenter Thm.
Substitute 13.4 for GB.

10. 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

11. 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.
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12. 5-2 Bisectors of Triangles
Unlike the circumcenter, the incenter is always inside
the triangle.
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13. 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

14. 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

15. 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.
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16. 5-2 Bisectors of Triangles
CW
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17. 5-2 Bisectors of Triangles
Objectives
Apply properties of medians of a
triangle.
Apply properties of altitudes of a
triangle.
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18. 5-2 Bisectors of Triangles
Vocabulary
median of a triangle
centroid of a triangle
altitude of a triangle
orthocenter of a triangle
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19. 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

20. 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

21. 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

22. 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.
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23. 5-2 Bisectors of Triangles
Helpful Hint
The height of a triangle is the length of an
altitude.
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24. 5-2 Bisectors of Triangles
vocabulary
The midsegment of a
triangle - Segment that
joins the midpoints of
any two sides of a
triangle.
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25. 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