Rhombus and diagonals

The a rhombus area is 150 cm2 and the ratio of the diagonals is 3:4. Calculate the length of its height.

Result

h =  12 cm

Solution:

$S=150 \ \\ u_{ 1 }:u_{ 2 }=3:4 \ \\ S=u_{ 1 } \ u_{ 2 } / 2 \ \\ u_{ 1 }=3k \ \\ u_{ 2 }=4k \ \\ S=3k \cdot \ 4k/2 \ \\ S=6 \ k^2 \ \\ k=\sqrt{ S/6 }=\sqrt{ 150/6 }=5 \ \\ u_{ 1 }=3k=3 \cdot \ 5=15 \ \\ u_{ 2 }=4k=4 \cdot \ 5=20 \ \\ a^2=(u_{ 1 }/2)^2+(u_{ 2 }/2)^2=(15/2)^2+(20/2)^2=\dfrac{ 625 }{ 4 }=156.25 \ \\ a=\sqrt{ (u_{ 1 }/2)^2+(u_{ 2 }/2)^2 }=\sqrt{ (15/2)^2+(20/2)^2 }=\dfrac{ 25 }{ 2 }=12.5 \ \\ S=ah \ \\ h=S/a=150/12.5=12 \ \text{cm}$

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