Question

An equilateral triangle of side \( 4\sqrt{3} \) 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. \( 2\sqrt{3} \)
• B. \( 4\sqrt{3} \)
• C. 12
• D. \( 8\sqrt{3} \)

Answer 1

The Correct Option is D

To find the least perimeter of the rectangle formed without loss of area from the equilateral triangle, we proceed as follows:

Firstly, calculate the area of the equilateral triangle. For a triangle with side \( s \), the area \( A \) is given by:

\( A = \frac{\sqrt{3}}{4} \cdot s^2 \)

Given \( s = 4\sqrt{3} \), substituting the value:

\( A = \frac{\sqrt{3}}{4} \cdot (4\sqrt{3})^2 = \frac{\sqrt{3}}{4} \cdot (16 \times 3) = 12\sqrt{3} \) cm²

Next, consider a rectangle with length \( l \) and breadth \( b \). The area of the rectangle is equal to the area of the triangle:

\( lb = 12\sqrt{3} \)

The perimeter \( P \) of the rectangle is given as:

\( P = 2(l + b) \)

To minimize the perimeter for a given area, the rectangle should be as close to a square as possible. This implies \( l = b \).

Setting \( l = b = x \), we get:

\( x^2 = 12\sqrt{3} \)

Thus,

\( x = \sqrt{12\sqrt{3}} \)

Calculating:

\( x = \sqrt{12} \cdot \sqrt[4]{3} = 2\sqrt{3} \) cm

Thus, the least perimeter becomes:

\( P = 2(2\sqrt{3} + 2\sqrt{3}) = 8\sqrt{3} \) cm

The least perimeter of the rectangle is \( 8\sqrt{3} \) cm.

Answer 2

Calculate the area of the equilateral triangle. The formula for the area of an equilateral triangle with side $a$ is:
Area $= \frac{\sqrt{3}}{4} \cdot a^2$
Substitute $a = 4\sqrt{3}$:
Area $= \frac{\sqrt{3}}{4} \cdot (4\sqrt{3})^2 = \frac{\sqrt{3}}{4} \cdot 48 = 12\sqrt{3} \, cm^2$
Dimensions of the rectangle. The rectangle has the same area as the triangle. Let the dimensions of the rectangle be $l$ (length) and $b$ (breadth). Then:
$l \cdot b = 12\sqrt{3}$
For the rectangle to have the least perimeter, it should be as close to a square as possible (to minimize $l + b$). Hence, let:
$l = b$
Then:
$l^2 = 12\sqrt{3} \implies l = \sqrt{12\sqrt{3}} = 2\sqrt[4]{12} \approx 2\sqrt{3} \, cm$
Thus, the dimensions are:
$l = b = 2\sqrt{3} \, cm$
Perimeter of the rectangle. The perimeter of a rectangle is given by:
Perimeter $= 2(l + b)$
Substitute $l = b = 2\sqrt{3}$:
Perimeter $= 2(2\sqrt{3} + 2\sqrt{3}) = 2 \cdot 4\sqrt{3} = 8\sqrt{3} \, cm$
Final Answer: The least perimeter of the rectangle is:
$8\sqrt{3}$