Diamond diagonals

Find the diamond diagonal's lengths if the area is 156 cm² and side is 13 cm long.

Correct result:

u =  21.6333 cm
v =  14.4222 cm

Solution:

$$\begin{gathered} S=156{\text{cm}}^2 \\ a=13\text{ cm } \\ S=ah={\displaystyle \frac{uv}{2}} \\ h=S\mathrm{/}a=156\mathrm{/}13=12\text{ cm } \\ (u\mathrm{/}2{)}^2+(v\mathrm{/}2{)}^2=a^2 \\ {u}^2+v^2=4a^2 \\ {u}^2+v^2=1352 \\ uv=2S=312 \\ {u}^2+(2S\mathrm{/}u{)}^2=4a^2 \\ {u}^2+312^2\mathrm{/}{u}^2=676 \\ x=u^2 \\ x+97344\mathrm{/}x=676 \\ {x}^2+97344=676x \\ {x}^2-676x+97344=0 \\ a=1;b=-676;c=97344 \\ D=b^2-4ac=676^2-4\cdot 1\cdot 97344=67600 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{676\pm \sqrt{67600}}{2}} \\ {x}_{1,2}={\displaystyle \frac{676\pm 260}{2}} \\ {x}_{1,2}=338\pm 130 \\ {x}_1=468 \\ {x}_2=208 \\ \text{ Factored form of the equation: } \\ (x-468)(x-208)=0 \\ {u}_1=\sqrt{x_1}=\sqrt{468}=6\sqrt{13}\doteq 21.6333 \\ {u}_2=\sqrt{x_2}=\sqrt{208}=4\sqrt{13}\doteq 14.4222 \\ u=u_1=21.6333=6\sqrt{13}=21.6333\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} v=2\cdot S\mathrm{/}u=2\cdot 156\mathrm{/}21.6333=4\sqrt{13}\doteq 14.4222=14.4222\text{ cm } \\ v=u_2 \\ {S}_2={\displaystyle \frac{u\cdot v}{2}}={\displaystyle \frac{21.6333\cdot 14.4222}{2}}=156{\text{cm}}^2 \\ {a}_2=\sqrt{(u\mathrm{/}2{)}^2+(v\mathrm{/}2{)}^2}=\sqrt{(21.6333\mathrm{/}2{)}^2+(14.4222\mathrm{/}2{)}^2}=13\text{ cm} \end{gathered}$$

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