Question : The length and breadth of a rectangle are 8 cm and 6 cm, respectively. The rectangle is cut on its four vertices such that the resulting figure is a regular octagon. What is the length of the side (in cm) of the octagon?
Option 1: $3\sqrt{11}-7$
Option 2: $5\sqrt{13}-8$
Option 3: $5\sqrt{7}-11$
Option 4: $6\sqrt{11}-9$
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Correct Answer: $3\sqrt{11}-7$
Solution :
Given: The length and breadth of a rectangle are 8 cm and 6 cm, respectively. The rectangle is cut on its four vertices such that the resulting figure is a regular octagon.
Let the sides of the octagon be $a$ cm.
From the figure, we get,
$a^2=\frac{(8–a)^2}{4}+\frac{(6–a)^2}{4}$
⇒ $a^2=16-4a+\frac{a^2}{4}+9-3a+\frac{a^2}{4}$
⇒ $a^2=\frac{a^2}{2}+25-7a$
⇒ $a^2+14a-50=0$
Solving this, we get,
$a=(3\sqrt{11}-7)$ cm
Hence, the correct answer is $3\sqrt{11}-7$.
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