The diamond

The diamond has an area S = 120 cm², and 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.

Final Answer:

a =  13 cm
h =  9.2308 cm

Step-by-step explanation:

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

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

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