Question

A square and an equilateral triangle have equal perimeters. If the diagonal of the square is \[6\sqrt{2}cm\] then the area of the triangle is.
(a) \[16\sqrt{2}\]\[c{{m}^{2}}\]
(b) \[16\sqrt{3}\]\[c{{m}^{2}}\]
(c) \[12\sqrt{2}\]\[c{{m}^{2}}\]
(d) \[12\sqrt{3}\]\[c{{m}^{2}}\]

Answer 1

Hint: In this question, from the given value of the diagonal of the square by using the formula we get the side length of the square. Then equate the perimeters of the triangle and square by using the respective formula to get the side of the equilateral triangle. Now, by substituting this value in the formula of the area of the equilateral triangle we get the result.

Complete step-by-step answer:
PERIMETER: Total length of the sides of a plane figure.
AREA: Space covered by a plane figure.
Let us assume the side length of the equilateral triangle as a and the side length of the square as s.
EQUILATERAL TRIANGLE: A triangle in which all the sides are equal is called an equilateral triangle.
Perimeter of an equilateral triangle of side length a is given by the formula \[3a\]
Area of an equilateral triangle of side length a is given by the formula
\[\dfrac{\sqrt{3}}{4}{{a}^{2}}\]
SQUARE: A rectangle is a square in which all four sides are equal. Hence, the diagonals are equal and bisect each other at right angles.
Perimeter of a square of side length s is given by the formula \[4s\]
Diagonal of a square of side length s is given by the formula \[\sqrt{2}s\]
Now, by comparing the given value of diagonal with the formula mentioned above we get,
\[\Rightarrow \sqrt{2}s=6\sqrt{2}\]
Let us now divide with a square root of 2 on both sides.
\[\therefore s=6cm\]
Now, as given in the question that the perimeters of the triangle and square are equal from the above formulae we get,
\[\Rightarrow 3a=4s\]
Now, by substituting the value of s in the above equation we get,
\[\Rightarrow 3a=4\times 6\]
Let us now divide with 3 on both the sides.
\[\Rightarrow a=4\times 2\]
Now, on further simplification we get,
\[\therefore a=8cm\]
Let us now substitute this value of a in the area of an equilateral triangle formula.
\[\Rightarrow \dfrac{\sqrt{3}}{4}{{a}^{2}}\]
Now, on substituting the value of a we get,
\[\Rightarrow \dfrac{\sqrt{3}}{4}{{\left( 8 \right)}^{2}}\]
Now, on further simplification we get,
\[\begin{align}
  & \Rightarrow \sqrt{3}\times 8\times 2 \\
 & \Rightarrow 16\sqrt{3}c{{m}^{2}} \\
\end{align}\]
Hence, the correct option is (b).

Note:It is important to note that we first need to calculate the side of the square as the value of the diagonal is given. We need to be careful while substituting the values in the formulae because substituting the values incorrectly or substituting in the wrong formulae changes the corresponding value and so the final result.

Instead of finding the value of side length of the square from the diagonal value we can first equate the perimeters. Then we get the relation between the side of the triangle and the side of the square. Now, on substituting this side of square in terms of side of triangle in the diagonal of square formula we can directly get the value of side of triangle which on further substitution gives the value of area.