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| Plane Geometry - Plane Figures (Geometric
Figures) - Triangles |
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Triangles |
Types of triangles
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Main
properties of triangles
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Congruence of triangles
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Theorems about congruence
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Similarity
of triangles, division of a line segment in a given ratio |
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Similarity criteria of triangles
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Area of a triangle
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Right-angled
Triangle
- The Pythagorean Theorem |
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The
Pythagorean theorem |
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Trigonometric functions of an acute angle defined in a right triangle
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Solving the right triangle |
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Equilateral
and Isosceles Triangle
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Equilateral
triangle
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Isosceles
triangle
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Oblique
or Scalene Triangle |
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Properties and rules
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Perpendicular bisectors,
triangles circumcenter
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Angle bisectors, the center of the triangle’s incircle
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The
median, the centroid of a triangle |
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The altitude of a triangle, orthocenter
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Triangle formulas |
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Similarity and congruence of
triangles use
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Congruence |
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| Types of triangles
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| Types of triangles categorized by their sides
are; a scalene triangle, isosceles triangle and
equilateral |
| triangle. |
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| The types of triangles categorized by their angles
are; an acute triangle, obtuse triangle and
right triangle. |
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| Main
properties of triangles
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| 1) A sum of triangle angles
a +
b
+ g
= 180°. |
| 2) Angles, lying
opposite the equal sides, are also equal, and inversely. |
| All angles in
an equilateral triangle are also equal. It follows, that each angle in an
equilateral triangle is equal |
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to 60 degrees.
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3) In any triangle, if one
side is extended, the exterior angle is equal to a sum of interior
angles, not
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supplementary.
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| The
sum of exterior angles is 360°. |
| 4)
Any side of a triangle is less than a sum of two other sides and
greater than their difference. |
| 5)
An angle, lying opposite the greatest side, is also the greatest
angle, and inversely. |
| Proof:
By turning the side BC,
of the scalene triangle ABC
below, around the vertex C
by the angle g
into |
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direction of the side AC,
obtained is the isosceles triangle BCD
with equal angles on the base BD.
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| Its
lateral side CD
< AC.
Angle a <
b', as b'
is the exterior angle of the triangle ABD,
and b'
< b,
therefore |
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a <
b. Thus, proved is the above
statement.
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| Congruence of triangles
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| 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: |
| 1) If a pair of corresponding sides and the included angle
are equal SAS (Side-Angle-Side). |
| 2) If their corresponding sides
are equal SSS. |
| 3) 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'. |
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| Similarity of triangles,
division of a line segment in a given ratio
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| 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 are; Two triangles are
similar: |
| 1) If all their corresponding angles are
equal. |
| 2) If all their
sides are proportional. |
| 3) 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'. |
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| Division of a line segment
AB to
equal parts: Example:
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| Division of the line segment
AB
in a given ratio: Example:
AC
: BC
= 1 : 2 |
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| Area of a triangle
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| The area
A
of any triangle is equal to one-half the product of any base and corresponding height
h. |
| 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. |
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| Right-angled
Triangle
- The Pythagorean Theorem |
| The
Pythagorean theorem
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| 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). |
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| 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: |
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| In
the figure below shown are two geometric proofs of Pythagoras'
theorem which claims that the area of the |
| square of the
hypotenuse (the side opposite the right angle) is equal to the
sum of areas of the squares of |
| other two sides, i.e.,
c2
= a2
+ b2. |
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First proof shows that the area of the
biggest red square with the side
a +
b is equal to the sum of
four equal right triangles and the square of the hypotenuse
c, therefore |
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(a
+
b)2
= 4
· 1/2 · ab
+ c2 |
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a2
+
2ab
+ b2
= 2ab
+ c2 |
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a2+
b2
= c2 |
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| Second
proof shows that the area of the square of the hypotenuse
c
is equal to the sum
of the same four right triangles and the area of the small square with side a
- b, therefore |
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c2 = 4
· 1/2 · ab
+ (a
-
b)2 |
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c2 = 2ab
+ a2
-
2ab
+ b2 |
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c2 =
a2+
b2 |
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| Trigonometric functions of an acute angle defined in a right triangle
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| Trigonometric functions of an acute angle are defined in a right triangle as a ratio of its
sides. |
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| 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 parts 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. |
| There are four basic cases that can occur,
given |
| a) hypotenuse and
angle, c) hypotenuse and leg, |
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b) leg and
angle,
d) two legs. |
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| Equilateral
and Isosceles Triangle
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| Equilateral
triangle
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| Isosceles
triangle
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| Oblique
or Scalene Triangle |
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Properties and rules
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Perpendicular bisectors,
triangles circumcenter
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Angle bisectors, the center of the triangle’s incircle
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The
median, the centroid of a triangle |
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The altitude of a triangle, orthocenter
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Triangle formulas |
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Similarity and congruence of
triangles use |
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Congruence |
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| Properties and rules
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| The sum of the angles of a triangle
is a +
b
+ g
= 180°. |
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Any side of a triangle is shorter than the sum of other two
sides. |
| Circumcircle |
| A
perpendicular bisector of a triangle is a straight line
passing through the midpoint of a side and being |
| perpendicular
to it. |
| The
perpendicular bisectors of the 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. |
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| Incircle |
| 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. |
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| Median
and 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. |
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| Altitude
and orthocenter |
| The
altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. |
| The
altitudes intersect in the orthocenter of the triangle.
See the picture above. |
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| Triangle
formulas
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| 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
-semiperimeter. |
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| Similarity and congruence of
triangles use
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| Similarity
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| Two triangles (or two plane figures) are similar if they have
corresponding angles equal |
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a = a', b
= b', g
= g', |
| hence corresponding pairs of sides in
proportion. |
| If
k is the ratio of sides of two similar triangles, then
a'
= ka,
b'
= kb,
c'
= kc, |
| hence |
P'
= kP |
| and |
A'
= k2A |
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| Example:
Given triangle ABC
is divided by the angle a
bisector into two triangles ABD
and ADC,
as is |
| shown in the picture. |
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| By
use of the similarity it can be shown that |
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| Proof:
Through the point D
drawn is the line segment DE
parallel to the side c,
hence the triangles, ABC |
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and EDC
are similar. Therefore |
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| Example:
Find the value of x
of the triangle shown in the picture. |
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| Solution: |
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| Congruence
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| Two triangles are congruent if they have identical size and shape
so that they can be exactly superimposed. |
| Thus, two triangles are
congruent: |
| a) if a pair of corresponding sides and the included
angle are equal, |
| b) if their corresponding sides are
equal, |
| c) if a
pair of corresponding angles and the included side are equal. |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
