Lines in a triangle
Median
Altitude
Definition
A segment joining the vertex
to the midpoint of the
opposite side
The perpendicular drawn
from the vertex to the
opposite side.
Name of Point of Intersection
Centroid
or
Center of Gravity
Orthocenter
Circumcenter
Perpendicular Bisector
Bisector
The perpendicular to the
segment at its midpoint
The semi straight line that
divides an angle into 2 equal
parts
or
Center of circumscribed
Circle
Incenter
or
center of inscribed circle
1
I. Construct triangle ABC such that AB=4cm ; CÂB=40º ; C B̂ A=70º.
a) Construct [Cx) the bisector relative to AĈB , it cuts [AB] at S.
b) Construct [BH] the altitude relative to [AC].
Find the measure of the following angles and justify your answer.
AĈB
II.
;
BŜC
;
CĤB
;
H B̂ C
Construct an isosceles triangle ABC with vertex A.
Construct the remarkable lines from the vertex A.
What do you notice?
III. Construct any equilateral triangle LMN.
Construct all the remarkable lines from all the vertices.
What do you notice ?
IV.
Construct any acute triangle, then construct the bisectors of the three vertices. What do you notice
?
V.
Construct any acute triangle ,then construct the medians relative to the three sides; what do you
notice?
VI. Construct any acute triangle , then construct the perpendicular bisectors relative to the three sides;
what do you notice?
VII. Construct any acute triangle , then construct the altitude relative to the three sides; what do you
notice?
VIII.
2
There are three sides and three angles.
The three angles always add to 180°.
Scalene
Triangle
Isosceles
Triangle
- No equal sides
- Two equal sides
- No equal angles
- To equal angles
Equilateral
Triangle
- Three equal
sides
- Three equal
angles always 60°
Right Triangle
Right Isosceles
Triangle
- One right angle
- A triangle which
has a right angle
in it.
- Two equal sides
- Two other equal
angles always of
45°
3
In a triangle
Altitude
Is the perpendicular drawn from the vertex to the opposite side at a right angle?
(The line holding the opposite side)
Note: In a triangle , the three altitudes intersect at one point called “
Orthocenter”.
In an obtuse triangle , the orthocenter is outside the triangle.
Median
Is the segment joining the vertex to the midpoint of the opposite side?
Note : In a triangle, the three medians intersect at a point called “Centroid”.
Perpendicular Is the perpendicular to the segment at its mid-point?
bisector of a
Note : The perpendicular bisectors intersect at one point called
segment
“ Circumcenter” : center of circumscribed circle.
Angular
Bisector
Is a ray that divides the angle into 2 adjacent and congruent angles?
Note: The three angular bisectors intersect at a point “ Incenter”:center of the
inscribed circle
4
Centroid
Intersection of
Orthocenter
Intersection of
Circumcenter
Intersection of
( Center of the circumscribed
circle)
a triangle
Incenter
Intersection of
medians
in a triangle
altitudes
in a triangle
perpendicular bisectors
angular bisectors
in
in a triangle
( Center of the inscribed circle)
5
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