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Branch of Mathematics |
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Mathematics |
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ganithguru |
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ALGEBRA Series(special)
GEOMETRY Theoretical Geometry Constructions Applied geometry
TRIGONOMETRY Angles and measures Trigonometric Ratio Algebra of T-functions Properties of triangles Applied Trigonometry
CALCULUS Examples of different kinds of functions Graph of functions Limit of a function Continuity of a function Differential Calculus Integral Calculus Applied Calculus
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Geometry: Ganithguru welcomes you to learn about Geometry here. Geometry deals with points, line, figures formed by lines, and kudos it extends to solid figures and their study. We begin with Triangles Triangles can be classified according to the relative
lengths of their sides: In an equilateral triangle, all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length. In a scalene triangle, all sides and internal angles are different from one another. Equilateral Isosceles Scalene By internal angles Triangles can also be classified according to their internal angles, measured here in degrees. A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle. Right triangles
obey the Pythagorean theorem: The sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with
additional properties that make calculations involving them easier. Triangles that do not have an angle that measures 90° are
called oblique triangles. A triangle that has all interior angles measuring less
than 90° is an acute triangle or acute-angled triangle. A triangle that has one angle that measures more than 90°
is an obtuse triangle or obtuse-angled triangle. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the
same measure, all three sides have the same length, and therefore such
triangle is equilateral.
Right Obtuse Acute Oblique
The measures of the interior angles of a triangle in
Euclidean space always add up to 180 degrees. This allows determination of
the measure of the third angle of any triangle as soon as the measures of two
angles are known. An exterior angle of a triangle is an angle that is a
linear pair (and hence supplementary) to an interior angle. The measure of an
exterior angle of a triangle is equal to the sum of the measures of the two
interior angles that are not adjacent to it; this is the exterior angle
theorem. The sum of the measures of the three exterior angles (one for each
vertex) of any triangle is 360 degrees. Triangle inequality The sum of the lengths of any two sides of a triangle always exceeds the length of the third side, a principle known as the triangle inequality. Since the vertices of a triangle are assumed to be non
collinear, it is not possible for the sum of the length of two sides be equal
to the length of the third side. Two triangles that are under a correspondence are said to
be similar if every angle of one triangle has the same measure as the
corresponding angle in the other triangle and the corresponding sides have
lengths that are in the same proportion. A few basic
theorems about similar triangles:
Some sufficient
conditions for a pair of triangles to be congruent are: 1) SAS
Postulate: Two sides in a triangle have the same length as two sides in the
other triangle, and the included angles have the same measure. 2) ASA:
Two interior angles and the included side in a triangle have the same measure
and length, respectively, as those in the other triangle. (The included side
for a pair of angles is the side that is common to them.) 3) 4) 5) Hypotenuse-Leg
(HL) Theorem: The hypotenuse and a leg in a right triangle have the same
length as those in another right triangle. 6) Hypotenuse-Angle
Theorem: The hypotenuse and an acute angle in one right triangle have the
same length and measure, respectively, as those in the other right triangle.
This is just a particular case of the An important case: Side-Side-Angle (or Angle-Side-Side)
condition: If two sides and a corresponding non-included angle of a
triangle have the same length and measure, respectively, as those in another
triangle, then this is not sufficient to prove congruence; but if the angle
given is opposite to the longer side of the two sides, then the triangles are
congruent. The Hypotenuse-Leg Theorem is a particular case of this criterion.
The Side-Side-Angle condition does not by itself guarantee that the triangles
are congruent because one triangle could be obtuse-angled and the other
acute-angled. Using right triangles and the concept of similarity, the
trigonometric functions sine and cosine can be defined. These are functions
of an angle which are investigated in trigonometry. The Pythagorean Theorem A central theorem is the Pythagorean Theorem, which states
in any right triangle, the square of
the length of the hypotenuse equals the sum of the squares of the lengths of
the two other sides. If the hypotenuse has length c, and the legs have
lengths a and b, then the theorem states that
The converse is true: if the lengths of the sides of a
triangle satisfy the above equation, then the triangle has a right angle
opposite side c. Some other facts about right triangles: The acute angles of a right triangle are complementary.
If the legs of a right triangle have the same length, then
the angles opposite those legs have the same measure. Since these angles are
complementary, it follows that each measures 45 degrees. By the Pythagorean
theorem, the length of the hypotenuse is the length of a leg times √2. In a right triangle with acute angles measuring 30 and 60
degrees, the hypotenuse is twice the length of the shorter side, and the
longer side is equal to the length of the shorter side times √3 :
For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosine rule and sine rule). Points, lines and circles associated with a triangle: here are hundreds of different constructions that find a
special point associated with (and often inside) a triangle, satisfying some
unique property: see the references section for a catalogue of them. Often
they are constructed by finding three lines associated in a symmetrical way
with the three sides (or vertices) and then proving that the three lines meet
in a single point: an important tool for proving the existence of these is
Ceva's theorem, which gives a criterion for determining when three such lines
are concurrent. Similarly, lines associated with a triangle are often
constructed by proving that three symmetrically constructed points are
collinear: here Menelaus' theorem gives a useful general criterion. In this
section just a few of the most commonly-encountered constructions are
explained. The circumcenter is
the center of a circle passing through the three vertices of the triangle. A perpendicular bisector of a triangle is a straight line
passing through the midpoint of a side and being perpendicular to it, i.e.
forming a right angle with it. The three perpendicular bisectors meet in a
single point, the triangle's circumcenter; this point is the center of the
circumcircle, the circle passing through all three vertices. The diameter of this
circle can be found from the law of sines stated above. Thales' theorem implies that if the circumcenter is
located on one side of the triangle, then the opposite angle is a right one.
More is true: if the circumcenter is located inside the triangle, then the
triangle is acute; if the circumcenter is located outside the triangle, then
the triangle is obtuse. The intersection of the altitudes is the orthocenter.
The intersection of the angle bisectors finds the center
of the incircle. An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, and the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system. Excircles: An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
The intersection of the medians is the centroid. A 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 three medians intersect in a single point, the triangle's
centroid. The centroid of a stiff triangular object (cut out of a thin sheet
of uniform density) is also its center of gravity: the object can be balanced
on its centroid. The centroid cuts every median in the ratio 2:1, i.e. the
distance between a vertex and the centroid is twice the distance between the
centroid and the midpoint of the opposite side.
The midpoints of the three sides and the feet of the three
altitudes all lie on a single circle, the triangle's nine-point circle. The
remaining three points for which it is named are the midpoints of the portion
of altitude between the vertices and the orthocenter. The radius of the
nine-point circle is half that of the circumcircle. It touches the incircle
(at the Feuerbach point) and the three excircles.
The centroid (yellow), orthocenter (blue), circumcenter
(green) and barycenter of the nine-point circle (red point) all lie on a
single line, known as Euler's line (red line). The center of the nine-point
circle lies at the midpoint between the orthocenter and the circumcenter, and
the distance between the centroid and the circumcenter is half that between
the centroid and the orthocenter. The center of the incircle is not in general located on
Euler's line. If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedian intersect in a single point, the symmedian point of the triangle. Computing the area
of a triangle Calculating the area of a triangle is an elementary
problem encountered often in many different situations. The best known and
simplest formula is:
Where S is area, b is the length of the base of the
triangle, and h is the height or altitude of the triangle. The term 'base'
denotes any side and 'height' denotes the length of a perpendicular from the
point opposite the side onto the side itself. Although simple, this formula is only useful if the height can be readily found. For example, the surveyor of a triangular field measures the length of each side, and can find the area from his results without having to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle Other Methods: Using trigonometry
The height of a triangle can be found through an
application of trigonometry. Using the labeling as in the image on the left,
the altitude is h = a sin γ. Substituting this in the formula S = ½bh
derived above, the area of the triangle can be expressed as:
(where α is the interior angle at A, β is the
interior angle at B, γ is the interior angle at C and c is the line AB). Furthermore, since sin α = sin (π - α) =
sin (β + γ), and similarly for the other two angles:
Using coordinates If vertex A is located at the origin (0, 0) of a Cartesian
coordinate system and the coordinates of the other two vertices are given by
B = (xB, yB) and C = (xC, yC), then the area S can be
computed as ½ times the absolute value of the determinant
For three general vertices, the equation is:
In three dimensions, the area of a general triangle {A =
(xA, yA, zA), B = (xB, yB,
zB) and C = (xC, yC, zC)} is the
Pythagorean sum of the areas of the respective projections on the three
principal planes (i.e. x = 0, y = 0 and z = 0):
Using Heron's
formula
The shape of the triangle is determined by the lengths of
the sides alone. Therefore the area S also can be derived from the lengths of
the sides. By Heron's formula: Three equivalent ways of writing Heron's
formula are
The area may also be computed by: S = r × s, where r is the in radius, and s is the semi perimeter. Computing the sides
and angles In general, there are various accepted methods of
calculating the length of a side or the size of an angle. Whilst certain
methods may be suited to calculating values of a right-angled triangle,
others may be required in more complex situations. Trigonometric ratios in right triangles Trigonometric functions A right triangle always includes a 90° (π/2 radians)
angle, here labeled C. Angles A and B may vary. Trigonometric functions
specify the relationships among side lengths and interior angles of a right
triangle. In right triangles, the trigonometric ratios of sine,
cosine and tangent can be used to find unknown angles and the lengths of
unknown sides. The sides of the triangle are known as follows: The hypotenuse is the side opposite the right angle, or
defined as the longest side of a right-angled triangle, in this case h. The opposite side is the side opposite to the angle we are
interested in, in this case a. The adjacent side is the side that is in contact with the
angle we are interested in and the right angle, hence its name. In this case
the adjacent side is b. [edit] Sine, cosine and tangent The sine of an angle is the ratio of the length of the
opposite side to the length of the hypotenuse. In our case
Note that this ratio does not depend on the particular
right triangle chosen, as long as it contains the angle A, since all those
triangles are similar. The cosine of an angle is the ratio of the length of the
adjacent side to the length of the hypotenuse. In our case
The tangent of an angle is the ratio of the length of the
opposite side to the length of the adjacent side. In our case
The acronym "SOHCAHTOA" is a useful mnemonic for
these ratios. Inverse functions The inverse trigonometric functions can be used to
calculate the internal angles for a right angled triangle with the length of
any two sides. Arcsine can be used to calculate an angle from the length
of the opposite side and the length of the hypotenuse.
Arccos can be used to calculate an angle from the length
of the adjacent side and the length of the hypontenuse.
Arctan can be used to calculate an angle from the length
of the opposite side and the length of the adjacent side.
In introductory geometry and trigonometry courses, the
notation sin−1, cos−1, etc., are often used in place of arcsine, Arccos,
etc. However, the arcsine, Arccos, etc., notation is standard in higher
mathematics where trigonometric functions are commonly raised to powers, as
this avoids confusion between multiplicative inverse and compositional
inverse. The sine and cosine rules Main articles: Law of sines and Law of cosines A triangle with sides of length a, b and c and angles of
α, β and γ respectively.
The law of sines, or sine rule, states that the ratio of
the length of a side to the sine of its corresponding opposite angle is
constant, that is
This ratio is equal to the diameter of the circumscribed
circle of the given triangle. Another interpretation of this theorem is that
every triangle with angles α, β and γ is similar to a triangle
with side lengths equal to sin α, sin β and sin γ. This
triangle can be constructed by first constructing a circle of diameter 1, and
inscribing in it two of the angles of the triangle. The length of the sides
of that triangle will be sin α, sin β and sin γ. The side
whose length is sin α is opposite to the angle whose measure is α,
etc. The law of cosines, or cosine rule, connects the length of
an unknown side of a triangle to the length of the other sides and the angle
opposite to the unknown side. As per the law: For a triangle with length of sides a, b, c and angles of
α, β, γ respectively, given two known lengths of a triangle a
and b, and the angle between the two known sides γ (or the angle
opposite to the unknown side c), to calculate the third side c, the following
formula can be used:
If the lengths of all three sides of any triangle are
known the three angles can be calculated:
Non-planar triangles A non-planar triangle is a triangle which is not contained
in a (flat) plane. Examples of non-planar triangles in non-Euclidean
geometries are spherical triangles in spherical geometry and hyperbolic
triangles in hyperbolic geometry. While the internal angles in planar triangles always add up to 180°, a hyperbolic triangle has angles that add up to less than 180°, and a spherical triangle has angles that add up to more than 180°. A hyperbolic triangle can be obtained by drawing on a negatively-curved surface, such as a saddle surface, and a spherical triangle can be obtained by drawing on a positively-curved surface such as a sphere. Thus, if one draws a giant triangle on the surface of the Earth, one will find that the sum of its angles is greater than 180°. It is possible to draw a triangle on a sphere such that each of its internal angles is equal to 90°, adding up to a total of 270°. ganithguru |
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I invite review and suggestions. All students can contact me at ganithguru for further guidance.
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Basic Facts:
An
altitude of a triangle is a straight line through a vertex and perpendicular
to (i.e. forming a right angle with) 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 length of
the altitude is the distance between the base and the vertex. The three
altitudes intersect in a single point, called the orthocenter of the
triangle. The orthocenter lies inside the triangle if and only if the
triangle is acute. The three vertices together with the orthocenter are said
to form an orthocentric system.
Nine-point
circle demonstrates symmetry where six points lie on the edge of the
triangle.
Euler's
line is a straight line through the centroid (orange), orthocenter (blue),
circumcenter (green) and center of the nine-point circle (red).
