Quadrilaterals

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__Quadrilaterals __
 From the latin root: //quadri// - four; //latus// - side  Definition: a closed four-sided figure. All of its sides must be straight. There are many different kinds of quadrilaterals, but all of them have several things in common. They all have four sides and four vertices, and their interior angles all add up to 360 degrees. We know many quadrilaterals by their special shapes and properties, like squares. Remember, if you see the word quadrilateral, it does not necessarily mean a figure with special properties like a square or rectangle! In word problems, be careful not to think that a quadrilateral has parallel sides or equal sides unless that is stated. =Regular Quadrilaterals: =

__Rectangle__ 
===  A rectangle is a quadrilateral with all of its interior angles equal to 90°. The opposite sides of a rectangle run parallel to each other. A rectangle can also go under the parallelogram catergory. A square is a type of rectangle. ===   <span style="font-family: 'Comic Sans MS',cursive;"> Area of a Rectangle <span style="color: rgb(0, 255, 169);"> Formula: A=lw <span style="color: rgb(165, 92, 250);"> Example:

<span style="color: rgb(31, 249, 169);"> a=3cm b=5cm

A=lw <span style="font-size: 130%; color: rgb(3, 3, 3);"><span style="font-size: 80%; color: rgb(70, 251, 176);"> =5x3 =15cm²

Therefore, the area of the rectangle is 15cm².

<span style="font-size: 120%; color: rgb(8, 7, 7);">Perimeter of a Rectangle Formula; P=2(l+w)

<span style="color: rgb(193, 112, 250);">Example:  <span style="color: rgb(193, 112, 250);">

<span style="color: rgb(35, 251, 170);">a=3cm b=5cm

P=2(l+w) =2(3+5) =2(8) =16cm

Therefore, the perimeter of the rectangle is 16cm. <span style="color: rgb(29, 135, 226);">**<span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(7, 8, 8);">   __  Squares  __ ** <span style="font-family: 'Comic Sans MS',cursive;">A square is a quadrilateral with all four of it's sides and angles equal. All of it's interior angles are 90°. Another quality of a square is that if you were to draw lines connecting opposet vertexes (corners) the diagonals would intercept at a 90° angle (or perpendicular). A square can be considered as a rectangle, parallelogram, or rhombus. <span style="font-family: 'Comic Sans MS',cursive;"> **<span style="color: rgb(5, 5, 5);">Area of a Square: **

Formula: A=s² <span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(255, 252, 0);">Example:

<span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(20, 190, 245);"><span style="font-size: 120%; color: rgb(5, 5, 5);">    <span style="color: rgb(61, 118, 179);"><span style="font-size: 110%; font-family: 'Comic Sans MS',cursive;">a=10cm

A=s² =10² =100cm²

Therefore, the area is 100cm². ** Perimeter of a square  ** : <span style="color: rgb(53, 149, 233);"> <span style="color: rgb(28, 107, 212);">Formula: P=4s <span style="color: rgb(248, 242, 48);">Example:   <span style="color: rgb(248, 242, 48);"> <span style="font-size: 110%; color: rgb(0, 183, 255); font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(22, 106, 197);"> a=10cm

P=4s =4x10 =40cm

Therefore, the perimeter is 40cm. <span style="font-family: 'Comic Sans MS',cursive;"> __** Parallelograms **__ <span style="font-family: 'Comic Sans MS',cursive;"> Opposite sides of a parallelogram are parallel and equal in length. The opposite angles are congruent. Angles "a" are the same, and angles "b" are the same. Squares, Rectangles and Rhombuses are all Parallelograms.

Any side of a parallelogram can be the base. Once you have the base, the height of the parallelogram is the length of any line segment perpendicular from the base parallel to the opposite side. Also, adjacent angels of a parallelogram always have a sum of 180 degrees. Each diagonal bisects one another.

The area of a parallelogram is A = b x h, "b" representing the length and "h" representing the height. Example: A=? b=10cm h=5cm A = b x h A = 10 x 5 A = 50cm² <span style="color: rgb(0, 0, 0);"><span style="background-color: rgb(255, 255, 255);">The area is 50 <span style="color: rgb(103, 15, 210); background-color: rgb(255, 255, 255);"><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: rgb(5, 5, 5);">cm²   __<span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(8, 7, 8); background-color: rgb(255, 255, 255);">. __<span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(8, 7, 8); background-color: rgb(255, 255, 255);"> <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(8, 7, 8); background-color: rgb(255, 255, 255);">

The perimeter of a parallelogram is the distance around the outside of it. The formula is side + side + side + side, since a parallelogram has four sides. With opposite sides being congruent, you could use the formula 2s+2s as long as you used two different sidelengths. Example: P = S+S+S+S P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm Therefore, the perimeter is 22cm.

__ Rhombus __ <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"><span style="color: rgb(151, 31, 168); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"> <span style="font-family: 'Comic Sans MS',cursive;"><span style="font-family: 'Lucida Console',Monaco,monospace;"><span style="font-family: 'Times New Roman',Times,serif;"><span style="font-family: 'Lucida Console',Monaco,monospace;"><span style="font-family: Tahoma,Geneva,sans-serif;">A rhombus is a different type of parallelogram. Instead of having opposite sides parallel and equal in length, all the sides of a rhombus have equal length. A rhombus has some of the same characteristics of a square but not all angles need to be 90°. All sides of a rhombus and be named the base because they are all equal       <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(151, 31, 168); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="font-family: 'Lucida Console',Monaco,monospace;"><span style="font-family: 'Times New Roman',Times,serif;"><span style="font-family: 'Lucida Console',Monaco,monospace;">. <span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(158, 37, 173); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">A=ba <span style="font-size: 80%; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"> <span style="color: rgb(149, 35, 159); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">A= base × altitude (altitude is also known as height) <span style="color: rgb(158, 37, 173); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"> <span style="font-family: 'Comic Sans MS',cursive;"> __<span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(0, 0, 0);"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);"> A trapezoid is a quadrilateral with one pair of parallel sides. The pair of parallel sides are not the same length, but the other two sides are. A trapezoid also has two obtuse angles, and two acute angles. <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);"> <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);">**<span style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);"> Area of a Trapezoid: ** <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">  Formula: A= h[(b1+b2)/2] <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(100, 15, 189); background-color: rgb(255, 255, 255);">Example: <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);"> <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">a= 10cm <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">b=9cm <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">c=5cm <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(103, 15, 210); background-color: rgb(255, 255, 255);"> <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(22, 243, 42);">height: 4cm
 * Area of a rhombus: **
 * <span style="background-color: rgb(246, 244, 244);"> <span style="font-family: 'Comic Sans MS',cursive;"> T rapezoids   **  __

<span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">A= h[(b1+b2)/2]   <span style="color: rgb(0, 255, 6);"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(19, 231, 54);">= 4[(10+9)/2] = 4(19/2) = 4x19.5 = 38cm²

Therefore, the area of this trapeziod is 38cm². <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(10, 10, 10);">** Perimeter of a Trapezoid: ** <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(16, 234, 32);">Formula: P= s+s+s+s Example:

<span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">

<span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">a= 10cm <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">b=9cm <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">c=5cm

<span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(0, 255, 6);"> P=s+s+s+s =10+9+5+5 =29cm

Therefore, the perimeter of the trapezoid is 29cm. <span style="font-family: 'Comic Sans MS',cursive;"> **__ Quadrilaterals - How Do They Relate To The Real World? __**

Many of our everyday items are quadrilaterals. Example. paper - rectangle, kite - rhombus, piece of cheese - square. :P Quadrilaterals are all around us so we need to know about them! <span style="font-family: 'Comic Sans MS',cursive;"> **__ Why Do We Need To Know The Perimeter of Quadrilaterals? __** There are some problems that we may come across, in which case we need to know how to solve them by learning about the perimeter of quadrilaterals.

<span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(242, 24, 24);">Example: Harley lives in a house with a square yard around it. Each side of her yard is 20m. She wants to build a fence around it, and her house (which is inside it). How much fencing will she need?

<span style="color: rgb(186, 69, 247);">-->To figure this out, we need to know the perimeter of her yard because that will tell us how much fencing she needs. To find the perimeter we need to know how to figure it out! Because we know the perimeter, this is how we find the answer: P=4s =4x20m =80m

Therefore, Harley needs 80m of fencing because the perimeter of her backyard is 80m. <span style="font-family: 'Comic Sans MS',cursive;">**__ Why Do We Need To Know The Area Of Quadrilaterals? __** <span style="font-size: 110%; color: rgb(246, 19, 19); font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(8, 8, 8);">In real life, there are lots of problems, and jobs, that deal with area. So, to figure them out, we need to know how! Example:

<span style="color: rgb(10, 10, 10);">Harley's house has a rectangular bathroom. The dimensions of her bathroom is 3m by 4m. Harley wants to cover the whole bathroom floor with tiles. How much tiles does she need? <span style="color: rgb(171, 0, 255);">-->To figure this out, we need to know the area of Harley's bathroom, because that will tell us how much tiling she will need to cover it. So, we use our area formula for a rectangle to get the answer: <span style="color: rgb(10, 10, 10);">A=lxw =3x4 =12m²

Therefore, Harley needs 12m² of tiling to cover her whole bathroom floor. <span style="display: block; font-size: 120%; color: rgb(5, 5, 5); font-family: Impact,Charcoal,sans-serif; text-align: center;">Irregular Quadrilaterals: <span style="display: block; color: rgb(248, 18, 18); font-family: 'Comic Sans MS',cursive; text-align: left;"> Irregular quadrilaterals still have four sides and four angles. Here are the differences between a regular and irregular quadrilateral:

<span style="display: block; color: rgb(248, 18, 18); font-family: 'Comic Sans MS',cursive; text-align: left;">REGULAR QUADRILATERALS IRREGULAR QUADRILATERALS - all four of its angles are convex (less than 180°) -can have concave angles (more than 180°) -have at least pme set of parallel sides <span style="display: block; font-size: 110%; color: rgb(246, 35, 35); font-family: 'Comic Sans MS',cursive; text-align: right;">- do not have any parallel sides <span style="display: block; font-size: 110%; color: rgb(245, 15, 15); font-family: 'Comic Sans MS',cursive; text-align: left;">-can have some or all of their side lengths the same - all of their side lengths can be different

<span style="color: rgb(104, 177, 27);">Here is an example of an irregular quadrilateral:

It is an irregular concave square. This is BECAUSE... <span style="color: rgb(97, 155, 70);">- it has a concave angle - it has no parallel sides


 * EVEN THOUGH SOME OF ITS SIDE LENGTHS ARE EQUAL, IT IS STILL IRREGULAR BECAUSE OF ITS OTHER ATTRIBUTES

<span style="color: rgb(0, 0, 0);">__ Some Other Pages That Might Help Your Understanding __:

Angles<span style="color: rgb(245, 15, 15);"> Composite Figures <span style="color: rgb(97, 155, 70);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(245, 15, 15);">Geometry    <span style="color: rgb(97, 155, 70);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(245, 15, 15);">Optimal ValuesParallel and Perpendicular Lines    <span style="color: rgb(97, 155, 70);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(245, 15, 15);">Polygons <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);"> <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="background-color: rgb(255, 255, 255);">  __ Citations **__ <span style="color: rgb(255, 127, 0);"><span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="color: rgb(171, 24, 180); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"><span style="font-size: 110%; color: rgb(249, 148, 31);"><span style="font-family: 'Comic Sans MS',cursive;">Page, John (2007) Rhombus. Math Open Reference 2007. Retrieved December 13, 2008 from: <span style="font-family: 'Comic Sans MS',cursive;">http://www.mathopenref.com/rhombus.html <span style="color: rgb(50, 242, 33); background-color: rgb(0, 255, 253);"><span style="color: rgb(255, 114, 0); background-color: rgb(255, 255, 255);"><span style="color: rgb(19, 12, 12);"> <span style="color: rgb(255, 127, 0);"><span style="color: rgb(0, 0, 0);"><span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(250, 144, 25);">Pierce, R. (n.d.). //Quadrilaterals//. Retrieved November 17, 2008, from  <span style="font-family: 'Comic Sans MS',cursive;">http://www.mathsisfun.com/quadrilaterals.html <span style="color: rgb(0, 0, 0);"><span style="font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(248, 136, 27);">Sanchez, P. (2007, July 18). //Parallelograms//. Retrieved December 16, 2008, from  <span style="font-family: 'Comic Sans MS',cursive;">http://planetmath.org/encyclopedia/Parallelogram.html  <span style="color: rgb(0, 0, 0);"><span style="color: rgb(16, 244, 20); background-color: rgb(255, 255, 255);"> <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="color: rgb(171, 24, 180); font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;"><span style="color: rgb(255, 127, 0);"><span style="color: rgb(0, 255, 14);"><span style="font-family: 'Comic Sans MS',cursive;"> <span style="color: rgb(253, 145, 23);"><span style="color: rgb(15, 245, 35);"><span style="color: rgb(250, 149, 25);">Page, John. (2007) //Trapezoid.// Math Open Reference 2007. Retrieved December 17, 2008 from:   <span style="font-family: 'Comic Sans MS',cursive;">[|**http://www.mathopenref.com/trapezoid.html**] <span style="font-family: 'Comic Sans MS',cursive;">Page, John. (2007) //Square//. Math Open Reference 2007. Retrieved December 17, 2008 from: http://www.mathopenref.com/square.html <span style="font-size: 110%; font-family: 'Comic Sans MS',cursive;"><span style="color: rgb(244, 136, 42);">Weisstein, Eric W.(1999-2008) //Square//. Wolfram MathWorld. Retrieved December 17, 2008 from : <span style="color: rgb(255, 128, 0);"><span style="font-size: 110%; font-family: 'Comic Sans MS',cursive;">http://mathworld.wolfram.com/Square.html <span style="font-size: 110%; font-family: 'Comic Sans MS',cursive;">Page. John. (2007). //Quadrilaterals//. Math Open Reference 2007. Retrieved December 18, 2008 from: <span style="font-size: 110%; font-family: 'Comic Sans MS',cursive;">http://www.mathopenref.com/quadrilateral.html Page, John. (2007). //Irregular Polygons//. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/polygonirregular.html Page, John. (2007). //Perimeter of a Rectangle//. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/rectangleperimeter.html Page, John. (2007). Rectangle. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/rectangle.html

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