Transcript TRIANGLES!!

TRIANGLES!!
Today, you will learn about various characteristics
and properties of triangles … so get your pencils
ready because there are A LOT of definitions!
Bisectors of a Triangle
Perpendicular bisector of
a triangle: a
line/ray/segment that
is perpendicular to a
side of the triangle at
the midpoint of the
side.
Perpendicular Bisectors of
different triangles
Concurrency
When three or more lines intersect in the same
point, they are called concurrent lines.
The point of intersection of the lines is called
the point of concurrency.
Thm:Concurrency of
Perpendicular Bisectors of a
Triangle
The perpendicular
bisectors of a triangle
intersect at a point that
is equidistant from the
vertices of a the
triangle.
*perpendicular bisectors
*distance from circumcenter to
vertex (all congruent)
The point of concurrency of the
perpendicular bisectors of a triangle
is called the circumcenter.
According to our thm., AP = BP = CP.
(Circumcenter is equidistant from vertices)
A
P
B
C
Bisectors of a Triangle
Angle bisector of a
triangle: a
line/ray/segment that
bisects the angle of a
triangle.
Thm:Concurrency of Angle
Bisectors of a Triangle
The angle bisectors of a
triangle intersect at a
point that is
equidistant from the
sides of a the triangle.
*angle bisectors
*distance from incenter to sides (all
congruent)
The point of concurrency of angle
bisectors of a triangle is called the
incenter.
According to our thm., KM = JM = ML.
(Incenter is equidistant from sides)
K
M
L
J
Example
a. Assuming there is a point at
A,B, and C, which point is
the best spot to place a
sprinkler to water plants
located at X,Y, and Z?
Z
10 in.
C
A
b. If the sprinkler covers a
circular region with a radius
of 15 in., will water reach
all three plants?
X
Y
B
Medians of a triangle
Median of a triangle: a
segment that starts at
the vertex of a triangle
and ends at the
midpoint of the
opposite side.
Thm: Concurrency of Medians of
a Triangle
The medians of a triangle
intersect at a point such
that the distance from the
vertex to that point is twothirds of the length of the
entire median within the
triangle.
* Medians of the triangle
The point of concurrency of medians
of a triangle is called the centroid.
According to our thm., PM = (2/3)PB. (This
holds for all 3 medians.)
P
C
R
A
M
B
Q
Altitudes of a Triangle
Altitude of a triange: a
line that starts at the
vertex and ends at the
opposite side (OR line
that contains the
opposite side) such
that is is perpendicular
to the opposite side.
Altitudes of different triangles
Concurrency of Altitudes of a
Triangle
The altitudes of a triangle
intersect at a point called
the orthocenter.
* Altitudes of the triangle
Example
C is the centroid of
GHJ and CM = 8.
a. Find HM
G
b. Find CH
H
C
J
Example
Where is the orthocenter located in ABC if:
a. m of angle A = m of angle B = m of angle C
b. m of angle A = m of angle B = 45 degrees
c. m of angle A = 110 degrees
Checkpoint!!
1. What is an altitude?
2. What is the pt of concurrency of altitudes?
3. What is a perpendicular bisector?
4. What is the pt of concurrency of per. bis.?
5. What is a median?
6. What is the pt of concurrency of medians?
7. What is an angle bisector?
8. What is the pt of concurrency of angle bisectors?