# The sides 2

The sides of a trapezoid are in the ratio 2:5:8:5. The trapezoid’s area is 245. Find the height and the perimeter of the trapezoid.

Result

h =  14
o =  70

#### Solution:

$a:b:c:d=2:5:8:5 \ \\ a=8x \ \\ b=d=5x \ \\ c=2x \ \\ a || c \ \\ \ \\ S=245 \ \\ \ \\ h=\sqrt{ d^2 - ((a-c)/2)^2 } \ \\ h=x \cdot \ \sqrt{ 5^2 - ((8-2)/2)^2 } \ \\ \ \\ h=x \cdot \ \sqrt{ 5^2 - ((8-2)/2)^2 } \ \\ h=x \cdot \ \sqrt{ 16 } \ \\ h=4x \ \\ \ \\ S=\dfrac{ a+c }{ 2 } h \ \\ S=\dfrac{ 8x+2x }{ 2 } h \ \\ S=\dfrac{ 8x+2x }{ 2 } \cdot \ 4x \ \\ S=5x \cdot \ 4x \ \\ x=\sqrt{ S/20 }=\sqrt{ 245/20 }=\dfrac{ 7 }{ 2 }=3.5 \ \\ \ \\ h=4 \cdot \ x=4 \cdot \ 3.5=14$
$a=8 \cdot \ x=8 \cdot \ 3.5=28 \ \\ b=5 \cdot \ x=5 \cdot \ 3.5=\dfrac{ 35 }{ 2 }=17.5 \ \\ d=5 \cdot \ x=5 \cdot \ 3.5=\dfrac{ 35 }{ 2 }=17.5 \ \\ c=2 \cdot \ x=2 \cdot \ 3.5=7 \ \\ \ \\ S_{1}=\dfrac{ a+c }{ 2 } \cdot \ h=\dfrac{ 28+7 }{ 2 } \cdot \ 14=245 \ \\ S_{1}=S \ \\ \ \\ o=a+b+c+d=28+17.5+7+17.5=70$

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