Question

If an equilateral triangle of area\[X\], and a square of area \[Y\] have the same perimeter, then\[X\] is:
A) Equal to \[Y\]
B) Greater than \[Y\]
C) Less than \[Y\]
D) Less than or equal to \[Y\]

Answer 1

Hint: The area of an equilateral triangle with each side of length \[a\] is \[\dfrac{{\sqrt 3 }}{4}{a^2}\] sq. cm.
The area of square with each side of length \[b\] is \[{b^2}\] sq. cm.
The perimeter of an equilateral triangle with each side of length \[a\] is \[3a\] cm.
The perimeter of the square with each side of length \[b\] is \[4b\] cm.

Complete step by step answer:
It is given that the area of the equilateral triangle is \[X\]. The area of the square is \[Y\]. They have the same perimeter.
To find the perimeter at first, we need to find the length of each side of the equilateral triangle and the square.
We know that the area of an equilateral triangle with each side of length \[a\] is \[\dfrac{{\sqrt 3 }}{4}{a^2}\] sq. cm.
Let us consider, each side of the equilateral triangle is \[a\].
Now we are going to find the value of each side of the equilateral triangle.
According to the problem,
\[\dfrac{{\sqrt 3 }}{4}{a^2} = X\]
Let us solve the above equation, we get,
\[{a^2} = \dfrac{{4X}}{{\sqrt 3 }}\]
On taking the square root, we get,
\[a = \dfrac{{2\sqrt X }}{{{3^{\dfrac{1}{4}}}}}\]
We know that the area of the square with each side of length \[b\] is \[{b^2}\] sq. cm.
Let us consider, each side of the square is \[b\].
Now we are going to find the value of each side of length in square.
According to the problem,
\[{b^2} = Y\]
On taking the square root, we get,
\[b = \sqrt Y \]
Now, on substituting “a” in the perimeter of the equilateral triangle, we get
Perimeter of the equilateral triangle= \[\dfrac{{6\sqrt X }}{{{3^{\dfrac{1}{4}}}}}\]
Now, on substituting “b” in the perimeter of the square, we get
The perimeter of the square is \[4\sqrt Y \].
As per the problem, the perimeter of the equilateral triangle and the square is equal. Then, we get,
\[\dfrac{{6\sqrt X }}{{{3^{\dfrac{1}{4}}}}} = 4\sqrt Y \]
Now let us square both sides, we get,
\[\dfrac{{36X}}{{\sqrt 3 }} = 16Y\]
On simplifying we get,
\[3\sqrt 3 X = 4Y\]
Let us simplify again, we get,
\[\dfrac{{3\sqrt 3 }}{4}X = Y\]
This is the relation between \[X,Y\].
This shows that \[X\] is less than \[Y\].

Hence, the correct option is C), \[X\] is less than\[Y\].

Note:
There is no value of \[X\] and \[Y\] such that \[X\] could be equal to \[Y\]. So, we cannot consider the option \[X\] is equal to\[Y\].