Rectangle - desc circle

The length of the sides of the rectangle is at a ratio of 1:3. The circle's radius circumscribed to the rectangle is 10 cm. Calculate the rectangle's perimeter.

Correct answer:

p =  50.6 cm

Step-by-step explanation:

$$\begin{gathered} a:b=1:3 \\ r=10\text{cm} \\ 3a=b \\ (2r{)}^2=a^2+b^2 \\ (2r{)}^2=a^2+(3a{)}^2 \\ 4r^2=10a^2 \\ a=\sqrt{4/10}\cdot r=\sqrt{4/10}\cdot 10=2\sqrt{10}\text{cm}\doteq 6.3246\text{cm} \\ b=3\cdot a=3\cdot 6.3246=6\sqrt{10}\text{cm}\doteq 18.9737\text{cm} \\ p=2\cdot (a+b)=2\cdot (6.3246+18.9737)=16\sqrt{10}\doteq 50.5964\text{cm} \\ \text{ Verifying Solution: } \\ k=a/b=6.3246/18.9737=\frac{1}{3}\doteq 0.3333 \\ d=\sqrt{a^2+b^2}=\sqrt{6.3246^2+18.9737^2}=20\text{cm} \end{gathered}$$

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