Perimeter and diagonal

The perimeter of the rectangle is 82 m, the length of its diagonal is 29 m. Find the dimensions of the rectangle.

Correct answer:

a =  21 m
b =  20 m

Step-by-step explanation:

$$\begin{gathered} o=82\text{ m } \\ u=29\text{ m } \\ {u}^2=a^2+b^2 \\ o=2(a+b) \\ {u}^2=a^2+(o\mathrm{/}2-a{)}^2 \\ 29^2=a^2+(82\mathrm{/}2-a{)}^2 \\ -2a^2+82a-840=0 \\ 2a^2-82a+840=0 \\ p=2;q=-82;r=840 \\ D=q^2-4pr=82^2-4\cdot 2\cdot 840=4 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{82\pm \sqrt{4}}{4}} \\ {a}_{1,2}={\displaystyle \frac{82\pm 2}{4}} \\ {a}_{1,2}=20.5\pm 0.5 \\ {a}_1=21 \\ {a}_2=20 \\ \text{ Factored form of the equation: } \\ 2(a-21)(a-20)=0 \\ a=a_1=21\text{ m} \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=a_2=20\text{ m}$

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