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=d2−((a−c)/2)2 h=x⋅ 52−((8−2)/2)2  h=x⋅ 52−((8−2)/2)2 h=x⋅ 16 h=4x  S=a+c2h S=8x+2x2h S=8x+2x2⋅ 4x S=5x⋅ 4x x=S/20=245/20=72=3.5  h=4⋅ x=4⋅ 3.5=14a: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⋅ x=8⋅ 3.5=28 b=5⋅ x=5⋅ 3.5=352=17.5 d=5⋅ x=5⋅ 3.5=352=17.5 c=2⋅ x=2⋅ 3.5=7  S1=a+c2⋅ h=28+72⋅ 14=245 S1=S  o=a+b+c+d=28+17.5+7+17.5=70a=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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