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Rational Triangle
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled (can have all sides multiplied by the same integer, namely a common multiple of their denominators) to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term Heronian triangle; SomosSomos, M., "Rational triangles", http://grail.eecs.csuohio.edu/~somos/rattri.html uses it to refer to triangles whose ratios of sides are rational; Conway and GuyConway, J. H., and Guy, R. K., "The only rational triangle", in ''The Book of Numbers'', 1996, Springer-Verlag, pp. 201 and 228–239. define a rational triangle as one with rational sides ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter ''s''. Triangles The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths ''a'', ''b'', and ''c'' is :s = \frac. Properties In any triangle, any vertex and the point where the opposite excircle touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has a length equal to the semiperimeter. If A, B, C, A', B', and C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (AA', BB', and CC', shown in red in the diagram) are known as splitters, and s = , AB, +, A'B, =, AB, +, AB' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through the midpoint of a given segment) and the ''angle bisector'' (a line that passes through the apex of an angle, that divides it into two equal angles). In three-dimensional space, bisection is usually done by a plane, also called the ''bisector'' or ''bisecting plane''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line, which meets the segment at its midpoint perpendicularly. The Horizontal intersector of a segment AB also has the property that each of its points X is equidistant from the segment's endpoints: (D)\quad , XA, = , XB, . The proof follows from and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, XB, ^2 \; . Property (D) is usually used for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Progression (mathematics)
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form : \frac,\ \frac,\ \frac,\ \frac, \cdots, where ''a'' is not zero and −''a''/''d'' is not a natural number, or a finite sequence of the form : \frac,\ \frac,\ \frac,\ \frac, \cdots,\ \frac, where ''a'' is not zero, ''k'' is a natural number and −''a''/''d'' is not a natural number or is greater than ''k''. Examples * 1, 1/2, 1/3, 1/4, 1/5, 1/6, sometimes referred to as the ''harmonic sequence'' * 12, 6, 4, 3, \tfrac, 2, … , \tfrac, … * 30, −30, −10, −6, − \tfrac, … , \tfrac * 10, 30, −30, −10, −6, − , … , \tfrac Sums of harmonic progressions Infinite harmonic progressions are not summable (sum to infinity). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a ''geometric series''. Elementary properties The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Such a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the n-th term of the sequence (a_n) is given by: :a_n = a + (n - 1)d, If there are ''m'' terms in the AP, then a_m represents the last term which is given by: :a_m = a + (m - 1)d. A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series. Sum Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states :c^2 = a^2 + b^2 - 2ab\cos\gamma, where denotes the angle contained between sides of lengths and and opposite the side of length . For the same figure, the other two relations are analogous: :a^2=b^2+c^2-2bc\cos\alpha, :b^2=a^2+c^2-2ac\cos\beta. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem: :c^2 = a^2 + b^2. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known. History Though the notion of the cosine was not yet developed in his time, Euclid's '' Elements'', dating back to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the ''radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sqrt. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ... or planar lamina, while ''surface area'' refers to the area of an open surface or the boundary (mathematics), boundary of a solid geometry, three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a plane curve, curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heron's Formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work ''Metrica'', though it was probably known centuries earlier. Example Let be the triangle with sides , and . This triangle’s semiperimeter is :s=\frac=\frac=16 and so the area is : \begin A &= \sqrt = \sqrt\\ &= \sqrt = \sqrt = 24. \end In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers. Alternate expressions Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways, :\begin A &=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
