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 side (in cm) of the octagon ?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 3√11 - 7.
Let ABCD be the rectangle such that AB = CD = 8 cm and BC = DA = 6 cm.
Let the side of the regular octagon = s
Let the 4 verticles be trimmed such that x is on sides BC and DA and y is on sides AB and CD.
So, at 4 verticles we are cutting off right angled triangles of sides x and y while their hypotenuse is s.
2y + s = 8 eq1
2x + s = 6 eq2
x² + y² = s² eq3
From eq1, y = (8 - s)/2
From eq2, x = (6 - s)/2
Now, eq3 becomes
[(6 - s)/2]² + [(8 - s)/2 ]² = s²
⇒ [(6 - s)/2]² + [(8 - s)/2 ]² = 4 s²
⇒ 36 - 12s + s² + 64 - 16s + s² = 4 s²
⇒ 100 - 28s - 2s² = 0
⇒ s² + 14s - 50 = 0
⇒ s = [-14 ± √(14² + 200)]/2
⇒ s = [-14 ± 6√11]/2
⇒ s = 3√11 - 7
So, the sides of the regular octagon are (3√11 - 7) cm