Difference between revisions of "Trapezoid"

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A '''trapezoid''' is a cool and pretty geometric figure that lies in a plane. It is also a type of [[quadrilateral]].
 
 
A '''trapezoid''' is a geometric figure that lies in a plane. It is also a type of [[quadrilateral]].
 
  
 
==Definition==
 
==Definition==
Trapezoids are characterized by having one pair of [[parallel]] sides.  In general it is probably safe to assume that "one pair" means "exactly one pair," so that [[parallelogram]]s are not also trapezoidsHowever, it is not clear that this is a universal [[mathematical convention]].
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Trapezoids are characterized by having one pair of [[parallel]] sides.  Notice that under this definition, every [[parallelogram]] is also a trapezoid(Careful: some authors insist that a trapezoid must have exactly one pair of parallel sides.)
  
 
==Terminology==
 
==Terminology==
The two parallel sides of the trapezoid are referred to as the ''bases'' of the trapezoid; the other two sides are called the ''legs''.  If the two legs of a trapezoid have equal [[length]], we say it is an [[isosceles trapezoid]]. All isosceles trapezoids are [[cyclic]].
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The two parallel sides of the trapezoid are referred to as the ''bases'' of the trapezoid; the other two sides are called the ''legs''.  If the two legs of a trapezoid have equal [[length]], we say it is an [[isosceles trapezoid]]. A trapezoid is cyclic if and only if it is isosceles.
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The median of a trapezoid is defined as the line connecting the midpoints of the two legs. Its length is the arithmetic mean of that of the two bases <math>\dfrac{b_1+b_2}{2}</math>. It is also parallel to the two bases.
  
Given any [[triangle]], a trapezoid can be formed by cutting the triangle with a cut parallel to one of the sides.  Similarly, given a trapezoid, one can reconstruct the triangle from which it was cut by extending the legs until they meet.   
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Given any [[triangle]], a trapezoid can be formed by cutting the triangle with a cut parallel to one of the sides.  Similarly, given a trapezoid that is not a parallelogram, one can reconstruct the triangle from which it was cut by extending the legs until they meet.   
  
 
==Related Formulas==
 
==Related Formulas==
*The [[area]] of the trapezoid is equal to the average of the bases times the height. <math>\dfrac{h(b_1+b_2)}{2}</math>
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If A denotes the area of a trapezoid, b_1,b_2 are the two bases, and the perpendicular height is h, we get
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A=\dfrac{h}{2}(b_1+b_2)
  
 
==See Also==
 
==See Also==
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*[[Parallel]]
 
*[[Parallel]]
  
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[[Category:Definition]]
 
[[Category:Geometry]]
 
[[Category:Geometry]]
[[Category:Definition]]
 

Revision as of 17:47, 24 April 2023

A trapezoid is a cool and pretty geometric figure that lies in a plane. It is also a type of quadrilateral.

Definition

Trapezoids are characterized by having one pair of parallel sides. Notice that under this definition, every parallelogram is also a trapezoid. (Careful: some authors insist that a trapezoid must have exactly one pair of parallel sides.)

Terminology

The two parallel sides of the trapezoid are referred to as the bases of the trapezoid; the other two sides are called the legs. If the two legs of a trapezoid have equal length, we say it is an isosceles trapezoid. A trapezoid is cyclic if and only if it is isosceles.

The median of a trapezoid is defined as the line connecting the midpoints of the two legs. Its length is the arithmetic mean of that of the two bases $\dfrac{b_1+b_2}{2}$. It is also parallel to the two bases.

Given any triangle, a trapezoid can be formed by cutting the triangle with a cut parallel to one of the sides. Similarly, given a trapezoid that is not a parallelogram, one can reconstruct the triangle from which it was cut by extending the legs until they meet.

Related Formulas

If A denotes the area of a trapezoid, b_1,b_2 are the two bases, and the perpendicular height is h, we get A=\dfrac{h}{2}(b_1+b_2)

See Also