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# A square is inscribed in a circle with radius 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 =

We know that diameter of a circle = d = 2 × r

∴ in the adjacent diagram, BD = 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}

∴ -------------------- (From equation i)

∴ 288 = (AD)^{2} + (AD)^{2 }.-------------------------------------. (Lenghths of the sides of a square are equal)

∴ 288 = 2(AD)^{2}

∴ = (AD)^{2}

∴ 144 = (AD)^{2}

∴ = 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 inches is 48 inches.

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