# 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.

Correct 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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