Perimeter

 Manipulative 1: Perimeter of a Triangle. Created with GeoGebra. 

The perimeter of a triangle is the sides
of the triangle or the sum of the lengths of the sides. For example, if the lengths
of the sides are 3, 4, and 5, the perimeter is 3 + 4 + 5 = 12. Click on the blue points
in manipulative 1 and drag them to change the figure.

Angle Sum Theorem

 Manipulative 2: Angle Sum Theorem. Created with GeoGebra. 

In Euclidean geometry, the
sum of the angles of a triangle is 180° = π radians. In other geometries, this might
not be true. Click on the blue points in manipulative 2 and drag them to change the figure.

Area

 Manipulative 3: Area of a triangle. Created with GeoGebra. 

The area of a triangle is
where
b (base) is any side of the triangle, and
h (height) is the distance from the vertex
opposite the base (in this case B) to the
extended base (in this case the line AC). Click on the blue points
in manipulative 3 and drag them to change the figure.

 Manipulative 4: Heron's formula for the area of a triangle. Created with GeoGebra. 

The area of a triangle can also be calculated from the
length of the three sides using Heron's Formula. First,
one must calculate the semiperimeter. This 1/2 of the
perimeter.
Since the perimeter is a + b + c where
a, b and
c are the length of the sides of the triangle,
the semiperimeter is .
Heron's formula for the area of a triangle is
.

Incircle Incenter

 Manipulative 5: Incircle and incenter of a triangle. Created with GeoGebra. 

The incircle
of a triangle is the circle that is tangent to each of the sides of a triangle. The
incenter is the center of the incircle. For more
information on the incenter of a triangle, see Incenter
from All Math Words Encyclopedia. Click on the blue points in the manipulative and
drag them to change the figure.

Circumcircle
Circumcenter

 Manipulative 6: Circumcircle and Circumcenter of a Triangle. Created with GeoGebra. 

The circumcircle of a triangle is the circle
that passes through all of the vertices of a triangle. The
circumcenter is the center of the circumcircle.
For more information on the circumcenter or circumcircle of a triangle, see
Circumcenter from All Math Words Encyclopedia.

Triangle Median Triangle Centroid

 Manipulative 7: Triangle centroid. Created with GeoGebra. 

A median of a triangle is a line drawn through
a vertex of the triangle and the midpoint of the opposite side. This means
that every triangle has three medians. The medians of a triangle meet at a
point called the centroid of the triangle.
The centroid
of a triangle is the center of gravity of the triangle. This means that if a
triangle is balanced on a pin at the centroid, it would be perfectly balanced.
The centroid of a triangle is found by drawing two
medians
of the triangle^{1}. The centroid is at the point where the medians
intersect.

Triangle Altitude

 Manipulative 8: Triangle altitude. Created with GeoGebra. 

An altitude of a triangle is a line segment from a
vertex
of the triangle to the extended opposite side, perpendicular to the opposite side.

Triangle Orthocenter

 Manipulative 9: Triangle orthocenter. Created with GeoGebra. 

The orthocenter
of a triangle is at the
intersection
of the
altitudes
of a triangle.

SAS Congruence

 Manipulative 10: Triangle SAS congruence. Created with GeoGebra. 

Two triangles
are congruent
if two adjacent
sides and the angle contained by the sides are congruent with
corresponding
sides and angle of the other triangle. In this case we say that the triangles are
SAS congruent.
SAS stands for side, angle, side.
For more information on SAS Congruence, see SAS Congruence from All Math Words Encyclopedia.

Proposition 6, Euclid's Elements: If two angles of a triangle are equal, the sides opposite the equal angles are also equal.

 Manipulative 11: Euclid proposition 6 SAS congruence. Created with GeoGebra. 

In a triangle, if two angles have equal length, the sides opposite the
equal angles are also equal. In figure 16, the angle ABC
is equal to the angle ACB. The side AB is
also equal to the side AC.
For more information on this property of triangles see:
