Trapezoid MO

The rectangular trapezoid ABCD with right angle at point B, |AC| = 12, |CD| = 8, diagonals are perpendicular to each other.

Calculate the perimeter and area of ​​the trapezoid.

Result

p =  33.31
A =  69.25

Solution:

∣AC∣=12 ∣CD∣=8  sin⁡Θ=∣BC∣∣AC∣ cos⁡Θ=∣BC∣∣BD∣  cos⁡2Θ+∣CD∣∣AC∣cos⁡Θ−1=0 x2+∣CD∣∣AC∣x−1=0  x2+0.667x−1=0  a=1;b=0.667;c=−1 D=b2−4ac=0.6672−4⋅1⋅(−1)=4.4444444444 D>0  x1,2=−b±D2a=−0.67±4.442 x1,2=−0.33333333±1.0540925533895 x1=0.72075922005613 x2=−1.3874258867228   Factored form of the equation:  (x−0.72075922005613)(x+1.3874258867228)=0  Θ=43∘52′58"  ∣BC∣=∣AC∣sin⁡Θ=8.3182260804446 ∣AB∣=∣AC∣cos⁡Θ=8.6491106406735 ∣AD∣=∣BC∣2+(∣AB∣−∣CD∣)2=8.3435142325775  o=∣AB∣+∣BC∣+∣CD∣+∣AD∣=33.31|AC| = 12 \ \\ |CD| = 8 \ \\ \ \\ \sin \Theta = \dfrac{|BC|}{|AC|} \ \\ \cos \Theta = \dfrac{|BC|}{|BD|} \ \\ \ \\ \cos^2 \Theta + \dfrac{ |CD|}{|AC|}\cos \Theta - 1 =0 \ \\ x^2 + \dfrac{ |CD|}{|AC|}x - 1 =0 \ \\ \ \\ x^2 +0.667x -1 =0 \ \\ \ \\ a=1; b=0.667; c=-1 \ \\ D = b^2 - 4ac = 0.667^2 - 4\cdot 1 \cdot (-1) = 4.4444444444 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ -0.67 \pm \sqrt{ 4.44 } }{ 2 } \ \\ x_{1,2} = -0.33333333 \pm 1.0540925533895 \ \\ x_{1} = 0.72075922005613 \ \\ x_{2} = -1.3874258867228 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -0.72075922005613) (x +1.3874258867228) = 0 \ \\ \ \\ \Theta = 43^\circ 52'58" \ \\ \ \\ |BC| = |AC| \sin \Theta = 8.3182260804446 \ \\ |AB| = |AC| \cos \Theta = 8.6491106406735 \ \\ |AD| = \sqrt{ |BC|^2 + (|AB|-|CD|)^2} = 8.3435142325775 \ \\ \ \\ o = |AB|+|BC|+|CD| + |AD| = 33.31
A=(∣AB∣+∣CD∣)⋅∣BC∣2=69.25A = \dfrac{(|AB|+|CD|)\cdot |BC|}{2}= 69.25

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