Plane
figures
|
Triangle
|
Types of triangles
classified by their sides
are; a scalene triangle, isosceles triangle and
equilateral triangle, or when
classified by angles as; an acute triangle, obtuse triangle and
right triangle.
|
Properties of triangles
|
a)
A sum of a triangle angles
a +
b
+ g
= 180°. |
b)
Angles, lying
opposite the equal sides, are also equal, and inversely. |
c)
In any triangle, if one
side is extended, the exterior angle is equal to a sum of interior
angles not supplementary.
|
|
d)
The
sum of exterior angles is 360°.
|
e)
Any side of a triangle is less than a sum of two other sides and
greater than their difference.
|
Proof:
By turning the side BC,
of the scalene triangle ABC
above, around the vertex C
by the angle g
into direction of the side AC,
obtained is the isosceles triangle BCD
with equal angles on the base BD. |
Its
lateral side CD
< AC.
Angle a <
b', as b'
is the exterior angle of the triangle ABD,
and b'
< b,
therefore a <
b. Thus, proved is the above
statement. |
|
Congruence of triangles
|
Two
figures are called congruent if they have identical size and shape, i.e., if their corresponding angles and sides are equal. |
The
two congruent figures fit on top of each other exactly. We prove the
congruence of two figures by rotation and translation. |
Theorems about congruence of triangles are; Two triangles are
congruent: |
a)
If a pair of corresponding sides and the included angle
are equal SAS (Side-Angle-Side). |
b)
If their corresponding sides
are equal SSS. |
c)
If a pair of corresponding angles and the included
side are equal ASA. |
The congruence of two triangles we denote as
D ABC
@
D A'B'C'. |
|
Similarity of triangles
|
Two plane figures are similar if differ in scale not in shape. Two
polygons are similar, if their angles are equal and sides are proportional. |
Similarity criteria of triangles
- Two triangles are
similar: |
a)
If all their corresponding angles are equal. |
b)
If all their
sides are proportional. |
c)
If one angle of a triangle is congruent to
one angle of another triangle and the sides that include those angles are proportional. |
Similarity of
two triangles is denoted as D
ABC ~
D A'B'C'. |
|
|
Right-angled
Triangle
|
The
Pythagorean theorem
|
In any right triangle the area of the square whose side is the
hypotenuse (the side of the triangle opposite the right angle) is equal
to the sum of the areas of the squares on the other two sides (legs). |
|
From the similarity of the triangles,
ADC,
BDC
and ABC,
and Thales’ theorem (an angle inscribed in
a semicircle is a right angle) proved is
Pythagoras’ theorem, |
|
|
Trigonometric functions of an acute angle defined in a right triangle
|
Trigonometric functions of an acute angle are defined in a right triangle as a ratio of its
sides. |
|
|
|
Solving
the right triangle
|
To solve a right triangle means to find all unknown sides and angles using its
known parts. |
While solving a right triangle we use
Pythagoras’ theorem and trigonometric functions of an
acute angle depending
which pair of its elements is given. |
Note, right triangles are usually denoted as
follows; c stands for the hypotenuse,
a
and b
for the perpendicular sides called legs, and a and
b
for the angles opposite to a
and b
respectively. |
|
Oblique
or scalene triangle
|
The triangle's circumcircle,
perpendicular bisectors
|
A
perpendicular bisector of a triangle is a straight line
passing through the midpoint of a side and being perpendicular
to it. Perpendicular bisectors of sides of any triangle are concurrent
(all pass through the same point). The perpendicular bisectors intersect in the
triangle's circumcenter. |
The triangle's circumcenter is the center of the
circumcircle which circumscribes given triangle passing through all its vertices. |
Acute
triangles' circumcenter falls inside the triangle. Obtuse
triangles' circumcenter falls outside the triangle. |
|
|
The triangle's incircle, the angle bisector
|
An
angle bisector is a straight line through a vertex of a
triangle that divides the angle into two equal parts. |
The three
angle bisectors intersect in a single point called the incenter,
the center of the triangle's incircle. |
Incircle
is a circle inscribed in a triangle so that each of the sides of
the triangle is a tangent, of which the radius is inradius,
therefore the radius is perpendicular from the incenter to any
side. |
|
The median of triangle,
centroid
|
The
median of a triangle is a straight line through a vertex
and the midpoint of the opposite side, and divides the triangle into
two equal areas. |
The medians intersect at the triangle's
centroid. |
The centroid cuts every median in the ratio
2 : 1 from a vertex to the midpoint of the opposite side. |
|
|
The
altitude of triangle, orthocenter
|
A height or
altitude of a triangle is a straight line through a vertex and perpendicular to the opposite
side. This opposite side is called the
base of the altitude, and the point where the altitude intersects the base (or its extension) is called
the foot of the altitude. |
The
altitudes intersect in the orthocenter of the triangle.
See the picture above. |
|
Triangle
formulas
|
Meaning
of symbols used in the above pictures and in the triangle
formulas are: |
h
-altitude, m
-median, t
-angle bisector,
r -radius
of the incircle, R
-radius of the
circumcircle, A
-area, |
P
-perimeter, s
-semi-perimeter. |
The area
A
of any triangle is equal to one-half the product of any base and corresponding height
h.
|
|
|
|
The
sine law
(or the sine rule) and the cosine law
|
From the congruence of triangles follows that an oblique triangle is
defined by three of its parts, as are |
- two sides and the included angle (SAS),
- two angles and the included side (ASA), |
- three sides (SSS)
and
- two sides and the angle opposite one of them
(SSA), |
which does not always
define a unique triangle. |
By using definitions of trigonometric functions of an acute angle and Pythagoras’ theorem, we can
examine mutual relationships of sides and corresponding angles of
a (an oblique) triangle. |
|
The
sine law
|
From
the right triangles,
BCD and
ACD
in the figure, |
hc
= a sinb
and hc
= b sina, |
so
that,
a
sinb
= b sina, |
|
|
|
|
Expressing
the same way the altitudes, hb and
ha as common legs of another
pairs of two right triangles in the above triangle, we get
|
hb
= a sing
and hb
= c sina, |
so
that,
a sing
= c sina, |
|
|
and,
ha
= b sing
and ha
= c sinb, |
so
that,
b
sing
= c sinb, |
|
|
|
These relations are called the
sine law and in words: Sides of a triangle are to one another in the same
ratio as sine of the corresponding (opposite) angles. |
|
The cosine law
|
|