Geometry Group 1

8/3/2019 Geometry Group 1

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Rectangle, Rhombus, Square, Trapezoid.

By:

Lisette Lao

Hanna Villareal

Katherine Thomas


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Quadrilaterals


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Rectangle


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A rectangle is

A plane figure with four straight sidesand four right angles, esp. one withunequal adjacent sides, in contrast to a

square.

*The word rectangle comes from theLatin "rectangulus", which is acombination of "rectus" (right) and"angulus" (angle)


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Formulas

If a rectangle has length (l) and width (w)

It has area

A = lw, It has perimeter

P = 2l + 2w = 2(l + w),


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Examples(Area)

Example 1:Find the area of a square witheach side measuring 2 inches.Solution: = (2 in) (2 in) = 4 in2

Example 2:A rectangle has a length of 8centimeters and a width of 3 centimeters.Find the area.

Solution: = (8 cm) (3 cm) = 24 cm2


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Examples(Area)

A rectangle is 4 times as long as it iswide. If the length is increased by 4inches and the width is decreased by 1

inch, the area will be 60 square inches.What were the dimensions of the originalrectangle?

- The dimensions of the original rectangleare 4 and 16.


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Examples(Perimeter)

Find the perimeter of a rectangular field oflength 45 m and width 35 m.

Solution:

So, the perimeter is 160 m.


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Examples(Perimeter)

Find the perimeter and area of a rectanglewith width 6 feet and length 14 feet.

Solution:

Length = 14 ftWidth = 6 ft

Perimeter of a rectangle = 2(14+6) = 2(20) =40 ft.Area of a rectangle = 14 * 6 = 84 squarefeet.


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Theorem 6.12

All angles of a rectangle are rightangles.


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Theorem 6.13

The diagonals of a rectangle areequal.


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Theorem 6.14If the diagonals of a

parallelogram are equal, then theparallelogram is a rectangle.


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Theorem 6.20

In a rectangle, the twodiagonals are congruent.


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Theorem 6.21

If an angle of a parallelogramis a right angle, then the

parallelogram is a rectangle.


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Theorem 6.22

In a quadrilateral, if thediagonals are congruent andbisect each other, then the

quadrilateral is a triangle.


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Rhombus


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Rhombus

Euclidian Geometry (def. of Rhombus):

a rhombus or rhomb is a convex

quadrilateral whose four sides all have thesame length.

A quadrilateral with all four sides equal inlength.

A parallelogram with a pair of adjacentsides equal.


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Characteristics of a Rhombus

A convex quadrilateral is a rhombus if and only if it isany one of the following parallelogram with four equalsides.

a parallelogram in which at least two consecutive

sides are congruent a quadrilateral with four congruent sides

a parallelogram in which a diagonal bisects an interiorangle

a parallelogram in which each diagonal bisects twointerior angles

a parallelogram in which the diagonals areperpendicular


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Perimeter and Area of a Rhombus

s = side length of rhombush = height of rhombusd1 = long diagonal of rhombusd2 = short diagonal of rhombus

Area = hs= s2 sin A= s2 sin B

= () d1d2

Perimeter = 4S


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Theorem 6.15

All sides of a rhombus are equal.


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Theorem 6.16

The diagonals of a rhombus areperpendicular to each other.


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Theorem 6.17

The diagonals of a rhombusbisect the angles at the vertices.


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Theorem 6.18

If the diagonals of a quadrilateralbisect each other at right angles,then it is a rhombus.


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Theorem 6.23

In a rhombus, the diagonals areperpendicular.


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Trapezoid


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Trapezoid

A trapezoid is a quadrilateral with twosides parallel. The trapezoid isequivalent to the British definition of

trapezium; An isosceles trapezoid is atrapezoid in which the base angles areequal so. A right trapezoid is a trapezoidhaving two right angles.

Isosceles trapezoid nonparallel sidesare equal.


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Area of a Trapezoid

or


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Perimeter of a Trapezoid

Perimeter = a + b + c + d

Where, a, b, c and d are the lengths ofeach side


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Theorem 6.21

Base angles of an isoscelestrapezoid are equal.


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Theorem 6.22

Diagonals of an isoscelestrapezoid are equal.


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Theorem 6.25Suppose the parallel lines L1 and L2

lie on the opposite sides of line L

such that L1 and L2 are equidistantfrom L. Let Rbe a point on L1 and Sa point on L2. then L bisects line RS.

Moreover, L is parallel to L1 and L2.


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Theorem 6.26

The median of a trapezoid isparallel to the two bases.


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Theorem 6.27

In an isosceles trapezoid, nointerior angle is a right angle.


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Theorem 6.28In an isosceles trapezoid ABCD

with AB = CD, the acute anglesformed by line AB and line CD

with line BC are congruent.


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Square


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What is a square?

A square is a regular quadrilateral. Thismeans that it has four equal sides andfour equal angles (90-degree angles, or

right angles). A square with verticesABCD would be denoted ABCD. Thesquare belong to the families of 2-

hypercube and 2-orthoplex.


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Perimeter and Area of a Square

Perimeter = 4L (length of sides)

Area = S2


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Properties

A square is a special case of a rhombus(equal sides), a kite (two pairs of adjacentequal sides), a parallelogram (oppositesides parallel), a quadrilateral or tetragon(four-sided polygon), and a rectangle(opposite sides equal, right-angles) andtherefore has all the properties of all theseshapes, namely:

The diagonals of a square bisect each

other and meet at 90 degrees. Thediagonals of a square bisect its angles.

The diagonals of a square areperpendicular.


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Opposite sides of a square areboth parallel and equal length.

All four angles of a square areequal. (Each is 360/4 = 90degrees, so every angle of asquare is a right angle.)

The diagonals of a square areequal.


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Additional Info

If the diagonals of a rhombus are equal, then thatrhombus must be a square. The diagonals of asquare are (about 1.414) times the length of aside of the square. This value, known asPythagoras constant, was the first number provento be irrational.

A square can also be defined as a rectangle withall sides equal, or a rhombus with all angles equal,or a parallelogram with equal diagonals that bisectthe angles.

If a figure is both a rectangle (right angles) and arhombus (equal edge lengths), then it is a square. If a circle is circumscribed around a square, the

area of the circle is / 2 (about 1.57) times thearea of the square.


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If a circle is inscribed in the square, the area of thecircle is / 4 (about 0.79) times the area of thesquare.

A square has a larger area than any otherquadrilateral with the same perimeter. A squaretiling is one of three regular tilings of the plane (theothers are the equilateral triangle and the regularhexagon).

The square is in two families of polytopes in twodimensions: hypercube and the cross polytope. TheSchlfli symbol for the square is {4}.

The square is a highly symmetric object. There arefour lines of reflectional symmetry and it hasrotational symmetry of order 4 (through 90, 180and 270). Its symmetry group is the dihedralgroup D4.


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Thats all. THANK YOU FORLISTENING!

Passed by: Lisette Lao,

Katherine Thomas & Hanna

Villareal

Documents

Geometry Group 1