Diamond diagonals

Calculate the diamond's diagonal lengths if its content is 156 cm² and the side length is 13 cm.

Correct answer:

u =  21.6333 cm
v =  14.4222 cm

Step-by-step explanation:

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

Our quadratic equation calculator calculates it.

$v=\sqrt{x_2}=\sqrt{208}=4\sqrt{13}=14.4222\text{ cm}$

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