A rectangle 2

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

Correct answer:

a =  63.4545 cm
b =  38.0727 cm

Step-by-step explanation:

$$\begin{gathered} d=74\text{ cm } \\ a:b=5:3 \\ {d}^2=a^2+b^2 \\ b={\displaystyle \frac{3}{5}}a \\ {d}^2=a^2+({\displaystyle \frac{3}{5}}a{)}^2 \\ {a}^2=d^2\mathrm{/}(1+{\displaystyle \frac{3^2}{5^2}}) \\ a={\displaystyle \frac{d}{\sqrt{1+{\displaystyle \frac{3^2}{5^2}}}}}={\displaystyle \frac{74}{\sqrt{1+{\displaystyle \frac{3^2}{5^2}}}}}=63.4545\text{ cm} \end{gathered}$$

$$\begin{gathered} b={\displaystyle \frac{3}{5}}\cdot a={\displaystyle \frac{3}{5}}\cdot 63.4545\doteq 38.0727=38.0727\text{ cm } \\ k=a\mathrm{/}b=63.4545\mathrm{/}38.0727={\displaystyle \frac{5}{3}}\doteq 1.6667 \\ {d}_2=\sqrt{a^2+b^2}=\sqrt{63.4545^2+38.0727^2}=74\text{ cm} \end{gathered}$$

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