Quadrilaterals

Quadrilateral - basic concepts

$A,B,C,D$ - vertices
$a=BC, b=CD, c=AD, d=AB$ - sides
$α, β, γ, δ$ - interior angles
$α', β', γ', δ'$ - exterior angles
$d_1, d_2$ - diagonals

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$$P=a+b+c+d$$ $$α+β+γ+δ=360°$$ $$α'+β'+γ'+δ'=360°$$

Types of quadrilaterals

Trapezoid

$a=AB, b=CD$ - bases
$l_1, l_2$ - sides
$d_1=AC, d_2=BD$ - diagonals
$h$ - height
$M$ - midpoint of AD
$N$ - midpoint of BC
$m=MN$ - central median $MN∥AB$

$$m={a+b}/2$$ $$α+δ=180°$$ $$β+γ=180°$$ $$d_1^2+d_2^2=l_1^2+l_2^2+2ab$$

If $l_1=l_2$ then $ABCD$ is Isosceles triangle and $α = β, d_1 = d_2$

If $α = 90°$ then $ABCD$ is right triangle

If $ABCD$ is circumscribed, then $a+b=2l$

Parallelogram

$AB=CD, BC=AD$ - sides
$α=γ, β=δ$ - angles
$α+δ=180°$, $β+γ=180°$
$d_1=AC, d_2=BD$ - diagonals
$h_a, h_b$ - heights

$$d_1^2+d_2^2=2a^2+2b^2$$ $$P=2(a+b)$$ $$AO=OC$$ $$BO=OD$$

Rectangle

$$d=d_1=d_2$$ $$d=√{a_2+b_2}$$ $$P=2(a+b)$$

Rhombus

$d_1, d_2$ - diagonals, bisectors of the interior angles

$$d_1⊥d_2$$ $$P=4a$$

Square

$$d=a√2$$ $$P=4a$$

Deltoid

$$d_1⊥d_2$$

Calculation of quadrilaterals' area

Rectangle

$$S=ab$$

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Parallelogram

$$S=ah$$

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Square

$$S=a^2$$ $$S=d^2/2^2$$

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Rhombus

$$S=ah$$ $$S={d_1d_2}/2$$

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Trapezoid

$$S=m.h$$ $$S={{a+b}/2}h$$

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Inscribed quadrilaterals

$$S=√{(p-a)(p-b)(p-c)(p-d)}$$

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Circumscribed quadrilateral

$$S={P_nr}/2$$

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