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Question

Mathematics Question on Triangles

An equilateral triangle of side 434\sqrt{3} cm formed out of a sheet is converted into a rectangle such that there is no loss of the area of the triangle. Then the least perimeter of the rectangle (in cm) will be:

A

232\sqrt{3}

B

434\sqrt{3}

C

12

D

838\sqrt{3}

Answer

838\sqrt{3}

Explanation

Solution

Calculate the area of the equilateral triangle. The formula for the area of an equilateral triangle with side aa is:
Area =34a2= \frac{\sqrt{3}}{4} \cdot a^2
Substitute a=43a = 4\sqrt{3}:
Area =34(43)2=3448=123cm2= \frac{\sqrt{3}}{4} \cdot (4\sqrt{3})^2 = \frac{\sqrt{3}}{4} \cdot 48 = 12\sqrt{3} \, cm^2
Dimensions of the rectangle. The rectangle has the same area as the triangle. Let the dimensions of the rectangle be ll (length) and bb (breadth). Then:
lb=123l \cdot b = 12\sqrt{3}
For the rectangle to have the least perimeter, it should be as close to a square as possible (to minimize l+bl + b). Hence, let:
l=bl = b
Then:
l2=123    l=123=212423cml^2 = 12\sqrt{3} \implies l = \sqrt{12\sqrt{3}} = 2\sqrt[4]{12} \approx 2\sqrt{3} \, cm
Thus, the dimensions are:
l=b=23cml = b = 2\sqrt{3} \, cm
Perimeter of the rectangle. The perimeter of a rectangle is given by:
Perimeter =2(l+b)= 2(l + b)
Substitute l=b=23l = b = 2\sqrt{3}:
Perimeter =2(23+23)=243=83cm= 2(2\sqrt{3} + 2\sqrt{3}) = 2 \cdot 4\sqrt{3} = 8\sqrt{3} \, cm
Final Answer: The least perimeter of the rectangle is:
838\sqrt{3}