Rhombus

The rhombus with area 137 has one diagonal that is longer by 5 than the second one. Calculate the length of the diagonals and rhombus sides.

Correct answer:

u₁ =  14.24
u₂ =  19.24
a =  11.97

Step-by-step explanation:

$$\begin{gathered} S=u_1{u}_2\mathrm{/}2 \\ {u}_2=u_1+5 \\ {u}_1^2+5u_1-2\cdot 137=0 \\ {x}^2+5x-274=0 \\ a=1;b=5;c=-274 \\ D=b^2-4ac=5^2-4\cdot 1\cdot (-274)=1121 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{-5\pm \sqrt{1121}}{2}} \\ {x}_{1,2}=-2.5\pm 16.7406690428 \\ {x}_1=14.2406690428 \\ {x}_2=-19.2406690428 \\ \text{ Factored form of the equation: } \\ (x-14.2406690428)(x+19.2406690428)=0 \\ {u}_1>0 \\ {u}_1=14.24 \end{gathered}$$

$u_2=u_1+5=19.24$

$$\begin{gathered} a^2=(u_1\mathrm{/}2{)}^2+(u_2\mathrm{/}2{)}^2 \\ a=\sqrt{(u_1\mathrm{/}2{)}^2+(u_2\mathrm{/}2{)}^2}=11.97 \end{gathered}$$

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