Parallelogram: Difference between revisions
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The area of a parallelogram can be seen as twice the area of a triangle created by one of its diagonals. The area of a parallelogram can be found by using the formula <math>B\times H=A</math> (Base multiplied by Height equals area). The area can also be computed as the magnitude of the [[vector cross product]] of two of its non-parallel sides. |
The area of a parallelogram can be seen as twice the area of a triangle created by one of its diagonals. The area of a parallelogram can be found by using the formula <math>B\times H=A</math> (Base multiplied by Height equals area). The area can also be computed as the magnitude of the [[vector cross product]] of two of its non-parallel sides. |
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==Proof that diagonals bisect each other== |
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[[Image:parallelogram.svg|right|Parallelogram ABCD]] |
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Prove that the diagonals of a parallelogram bisect each other. |
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(Prove that <math>E/\overrightarrow{A C}=1:1</math> and <math>E/\overrightarrow{B D}=1:1</math>) |
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Proof: |
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<math>\overrightarrow{A E}=k\overrightarrow{A C}</math>, k is an element of the real numbers |
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<math>\overrightarrow{A E}=k(\overrightarrow{A D}+\overrightarrow{D C}) |
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\Rightarrow |
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\overrightarrow{A E}=k(\overrightarrow{A D}+\overrightarrow{A B})</math> since <math>\overrightarrow{D C}=\overrightarrow{A B}</math> |
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<math>\overrightarrow{A E}=k\overrightarrow{A D}+k\overrightarrow{A B}</math> |
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since E,D,B are collinear, by the [[division-point theorem]], |
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k + k = 1 |
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2k = 1 |
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k = 0.5 |
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sub k = 0.5 into: |
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<math>\overrightarrow{A E}=k\overrightarrow{A C}</math> |
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<math>\overrightarrow{A E}=(0.5)\overrightarrow{A C}</math> |
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<math>\overrightarrow{A E} / \overrightarrow{A C}=0.5</math> (the ratio of AE to AC is 1:2) |
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also sub k = 0.5 into: |
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<math>\overrightarrow{A E}=k\overrightarrow{A D}+k\overrightarrow{A B}</math> |
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<math>\overrightarrow{A E}=0.5\overrightarrow{A D}+0.5\overrightarrow{A B}</math> |
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by the division-point theorem, |
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<math>E/\overrightarrow{D B}=1:1</math> |
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[[Image:Parallelogram2.PNG|right|Parallelogram ABCD]] |
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by adding the division ratios to the parallelogram, we see that E divides both diagonals in the ratio 1:1, and E bisects AC and BD. |
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Therefore, the diagonals of a parallelogram bisect each other. |
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== See also == |
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* [[Parallelogram of force]] |
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* [[Fundamental parallelogram]] |
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* [[Synthetic geometry]] |
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== External links == |
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* [http://mathworld.wolfram.com/Parallelogram.html Mathworld: Parallelogram] |
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* [http://www.cut-the-knot.org/Curriculum/Geometry/AreaOfParallelogram.shtml Area of Parallelogram] at [[cut-the-knot]] |
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* [http://www.cut-the-knot.org/Curriculum/Geometry/EquiTriOnPara.shtml Equilateral Triangles On Sides of a Parallelogram] at [[cut-the-knot]] |
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*[http://agutie.homestead.com/files/VarigWitten.htm Varignon and Wittenbauer Parallelograms] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas" |
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*[http://agutie.homestead.com/files/vanaubel.html Van Aubel's theorem] Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas" |
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* [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=2623&CurriculumID=24 Parallelogram Quiz] |
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* [http://www.mathopenref.com/parallelogram.html Definition and properties of a parallelogram] with animated applet |
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* [http://www.mathopenref.com/parallelogramarea.html Interactive applet showing parallelogram area calculation] interactive applet |
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[[Category:Quadrilaterals]] |
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[[bg:Успоредник]] |
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[[ca:Paral·lelogram]] |
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[[cs:Rovnoběžník]] |
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[[da:Parallelogram]] |
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[[de:Parallelogramm]] |
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[[es:Paralelogramo]] |
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[[fr:Parallélogramme]] |
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[[it:Parallelogramma]] |
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[[ko:평행사변형]] |
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[[he:מקבילית]] |
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[[lv:Paralelograms]] |
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[[hu:Paralelogramma]] |
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[[nl:Parallellogram]] |
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[[ja:平行四辺形]] |
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[[no:Parallellogram]] |
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[[pl:Równoległobok]] |
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[[pt:Paralelogramo]] |
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[[ru:Параллелограмм]] |
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[[fi:Suunnikas]] |
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[[sv:Parallellogram]] |
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[[ta:இணைகரம்]] |
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[[vi:Hình bình hành]] |
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[[zh:平行四边形]] |
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Revision as of 15:11, 5 November 2006
A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. Every parallelogram is a polygon, and more specifically a quadrilateral. Special cases of a parallelogram are the rhombus, in which all four sides are of equal length, the rectangle, in which the two sets of opposing, parallel sides are perpendicular to each other, and the square, in which all four sides are of equal length and the two sets of opposing, parallel sides are perpendicular to each other. In any parallelogram, the diagonals bisect each other, i.e, they cut each other in half.
The parallelogram law distinguishes Hilbert spaces from other Banach spaces.
It is possible to create a tessellation with any parallelogram.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
The area of a parallelogram can be seen as twice the area of a triangle created by one of its diagonals. The area of a parallelogram can be found by using the formula (Base multiplied by Height equals area). The area can also be computed as the magnitude of the vector cross product of two of its non-parallel sides.
