Question

The apothem of a square having its area numerically equal to its perimeter is compared to the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:
1) Equal to the second
2) $\dfrac{4}{3}$ times the second
3) $\dfrac{\sqrt{2}}{\sqrt{3}}$ times the second
4) $\dfrac{2}{\sqrt{3}}$ times the second
5) Indeterminately related to the second

Answer 1

Hint:
Here we have to find the relation between the apothem of a square and the apothem of an equilateral triangle. For that, we will use the relation between the side of a square and apothem of a square and then we will equate the numerical value of the area of the square with the perimeter of the square. From there, we will get the value of the apothem of the square. Then, we will use the relation between the side of an equilateral triangle and apothem of an equilateral triangle and then we will equate the numerical value of the area of the equilateral triangle with the perimeter of the equilateral triangle. From there, we will get the value of the apothem of the equilateral triangle. Then we will find the ratio of the apothem of square and the apothem of the equilateral triangle to get the required relation.

Complete step by step solution:
Let’s first consider the square.
Let each side of a square be $S_1$ and let the apothem of a square be $A_1$
Area of a square$={{S}_{1}}^{2}$
Perimeter of square$=4{{S}_{1}}$
According to the question, the numerical value of the area of the square is equal to its perimeter.
Therefore,
${{S}_{1}}^{2}=4{{S}_{1}}$
Subtracting $4{{S}_{1}}$from${{S}_{1}}^{2}$, we get
${{S}_{1}}^{2}-4{{S}_{1}}=0$
On further simplification, we get
$\begin{align}
  & {{S}_{1}}({{S}_{1}}-4)=0 \\
 & {{S}_{1}}=4\And {{S}_{1}}\ne 0 \\
\end{align}$
Thus, the side of a square is 4 units.
We know the relation between apothem of a square and its sides.
$2{{A}_{1}}={{S}_{1}}$
Putting the value of${{S}_{1}}$ , we get
$\begin{align}
  & 2{{A}_{1}}=4 \\
 & {{A}_{1}}=2 \\
\end{align}$
We got the value of apothem of a square.
Now, we will find the value of apothem of a given equilateral triangle.
Let each side of an equilateral triangle be $S_2$ and let the apothem of the equilateral triangle be $A_2$
Area of the equilateral triangle $ =\dfrac{\sqrt{3}}{4}{{S}_{2}}^{2}$
Perimeter of equilateral triangle $=3{{S}_{2}}$
According to the question, the numerical value of the area of an equilateral triangle is equal to its perimeter.
Therefore,
$\dfrac{\sqrt{3}}{4}{{S}_{2}}^{2}=3{{S}_{2}}$
Subtracting $3{{S}_{2}}$ from $\dfrac{\sqrt{3}}{4}{{S}_{2}}^{2}$, we get
$\dfrac{\sqrt{3}}{4}{{S}_{2}}^{2}-3{{S}_{2}}=0$
On further simplification, we get
$\begin{align}
  & {{S}_{2}}(\dfrac{\sqrt{3}}{4}{{S}_{2}}-3)=0 \\
 & {{S}_{2}}=4\sqrt{3}\And {{S}_{2}}\ne 0 \\
\end{align}$
Thus, the side of the equilateral triangle is $4\sqrt{3}$ units.
We know the relation between apothem of an equilateral triangle and its sides.
$2\sqrt{3}{{A}_{2}}={{S}_{2}}$
Putting the value of ${{S}_{2}}$, we get
$\begin{align}
  & 2\sqrt{3}{{A}_{2}}=4\sqrt{3} \\
 & {{A}_{2}}=2 \\
\end{align}$
We got the value of apothem of the equilateral triangle.
The apothem of square is equal to the apothem of the equilateral triangle.
${{A}_{1}}={{A}_{2}}$

Thus, the correct option is A.

Note:
Since we have calculated the apothem of square and apothem of equilateral triangle. Let’s define it to understand its meaning.
1) An apothem is defined as a length of the line segment from the midpoint of any of the sides of a regular polygon to the center of the regular polygon.
2) A regular polygon is defined as a plane shape having equal sides and equal angles. Example- square, equilateral triangle etc.