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Maths-

General

Easy

Question

A square is inscribed in a circle with radius $6 square root of 2$ inches. What is the perimeter of the square in inches?

The correct answer is: 12

Diagonal of the square inscribed in a circle = the diameter of the circle (refer diagram).
Perimeter of a square = 4 × Length of its Side
Pythagoras theorem: (Hypoteneous)² = sum of the squares of the remaining 2 sides.
Step-by-step solution:-
We are given that radius of the circle = r = $6 space square root of 2$
We know that diameter of a circle = d = 2 × r
$equals 2 cross times 6 square root of 2$
$therefore d equals 12 square root of 2$
∴ in the adjacent diagram, BD = $12 square root of 2$ units .------------------. (equation i)
Now, we know that adjacent sides of a square are perpendicular to each other, forming a right angle between the 2.
Hence, forming an equilateral triangle ABD in the adjacent figure.
Now, applying pythagoras theorem, we get-
(Hypoteneous)² = Sum of the squares of the remaining sides
∴ (BD)² = (AD)² + (BD)²
∴ $left parenthesis 12 square root of 2 right parenthesis squared equals left parenthesis AD right parenthesis squared plus left parenthesis BD right parenthesis squared$ -------------------- (From equation i)
∴ 288 = (AD)² + (AD)².-------------------------------------. (Lenghths of the sides of a square are equal)
∴ 288 = 2(AD)²
∴ $288 over 2$ = (AD)²
∴ 144 = (AD)²
∴ $square root of 144$ = AD ---------------------- (Taking square roots both the sides)
∴ ± 12 = AD
Since length of a side of a square cannot be negative,
∴ AD = 12 -------------- (equation ii)
Now, we know that perimeter of a square = 4 × length of side
∴ perimeter of ABCD = 4 × AD
= 4 × 12 --------------- (From ii)
= 48 inches
Final Answer:-
∴ Perimeter of a square inscribed in a circle with radius $6 square root of 2$ inches is 48 inches.

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