The diamond

The diamond has an area S = 120 cm², the ratio of the length of its diagonals is e: f = 5: 12. Find the lengths of the side and the height of this diamond.

Correct answer:

a =  13 cm
h =  9.2308 cm

Step-by-step explanation:

$$\begin{gathered} S=120{\text{cm}}^2 \\ e:f=5:12 \\ e=5x \\ f=12x \\ S={\displaystyle \frac{e\cdot f}{2}} \\ S={\displaystyle \frac{5x\cdot 12x}{2}} \\ x=\sqrt{{\displaystyle \frac{2\cdot S}{5\cdot 12}}}=\sqrt{{\displaystyle \frac{2\cdot 120}{5\cdot 12}}}=2\text{ cm } \\ e=5\cdot x=5\cdot 2=10\text{ cm } \\ f=12\cdot x=12\cdot 2=24\text{ cm } \\ a=\sqrt{(e\mathrm{/}2{)}^2+(f\mathrm{/}2{)}^2}=\sqrt{(10\mathrm{/}2{)}^2+(24\mathrm{/}2{)}^2}=13\text{ cm} \end{gathered}$$

$$\begin{gathered} S=a\cdot h \\ h=S\mathrm{/}a=120\mathrm{/}13={\displaystyle \frac{120}{13}}\doteq 9.2308=9.2308\text{ cm } \\ \text{ Verifying Solution: } \\ {S}_1={\displaystyle \frac{e\cdot f}{2}}={\displaystyle \frac{10\cdot 24}{2}}=120 \end{gathered}$$

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