# A rectangle 2

A rectangle has a diagonal length of 74cm. Its side lengths are in ratio 5:3. Find its side lengths.

Correct result:

a =  63.454 cm
b =  38.072 cm

#### Solution:

$d=74 \ \text{cm} \ \\ a:b=5:3 \ \\ \ \\ d^2=a^2 + b^2 \ \\ b=\dfrac{ 3 }{ 5 } a \ \\ d^2=a^2 + (\dfrac{ 3 }{ 5 } a)^2 \ \\ \ \\ a^2=d^2 / (1 + \dfrac{ 3^2 }{ 5^2 } ) \ \\ \ \\ a=\dfrac{ d }{ \sqrt{ 1 + \dfrac{ 3^2 }{ 5^2 } } }=\dfrac{ 74 }{ \sqrt{ 1 + \dfrac{ 3^2 }{ 5^2 } } }=63.454 \ \text{cm}$
$b=\dfrac{ 3 }{ 5 } \cdot \ a=\dfrac{ 3 }{ 5 } \cdot \ 63.4545=38.0724=38.072 \ \text{cm} \ \\ \ \\ k=a/b=63.4545/38.0724 \doteq \dfrac{ 5 }{ 3 } \doteq 1.6667 \ \\ \ \\ d_{2}=\sqrt{ a^2+b^2 }=\sqrt{ 63.4545^2+38.0724^2 } \doteq 73.9994 \ \text{cm}$

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