Rhombus and diagonals

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

Correct answer:

h =  12 cm

Step-by-step explanation:

$$\begin{gathered} S=150{\text{cm}}^2 \\ {u}_1:u_2=3:4 \\ S=u_1{u}_2\mathrm{/}2 \\ {u}_1=3k \\ {u}_2=4k \\ S=3k\cdot 4k\mathrm{/}2 \\ S=6k^2 \\ k=\sqrt{S\mathrm{/}6}=\sqrt{150\mathrm{/}6}=5\text{ cm } \\ {u}_1=3k=3\cdot 5=15\text{ cm } \\ {u}_2=4k=4\cdot 5=20\text{ cm } \\ {a}^2=(u_1\mathrm{/}2{)}^2+(u_2\mathrm{/}2{)}^2 \\ a=\sqrt{(u_1\mathrm{/}2{)}^2+(u_2\mathrm{/}2{)}^2}=\sqrt{(15\mathrm{/}2{)}^2+(20\mathrm{/}2{)}^2}={\displaystyle \frac{25}{2}}=12.5\text{ cm } \\ S=ah \\ h=S\mathrm{/}a=150\mathrm{/}12.5=12\text{ cm} \end{gathered}$$

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