# A rectangle 2

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

Result

a =  63.454 cm
b =  38.072 cm

#### Solution:

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

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