Question

The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:
A. equal to the second
B. $\dfrac{4}{3}$ times the second
C. $\dfrac{{\sqrt 2 }}{{\sqrt 3 }}$ times the second
D. $\dfrac{2}{{\sqrt 3 }}$ times the second
E. indeterminately related to the second

Answer 1

Hint: First we need to find the length of sides for the square and the equilateral triangle for the given conditions. Then we have to calculate and compare the apothems of both the square and the equilateral triangle.

Complete answer:
We are given that the apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter.

Apothem means –
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the centre to the midpoint of one of its sides. Equivalently, it is the line drawn from the centre of the polygon that is perpendicular to one of its sides.
The word "apothem" can also refer to the length of that line segment.

Firstly we find the length of the sides of the square for the given conditions. Therefore,
Let us assume that the length of the side is$x$.
Area = Perimeter (Given) … (1)
For the square with sides area is ${x^2}$ and perimeter is $4x$.
From equation (1) we have,
${x^2} = 4x$
Dividing both sides by $x$, we have,
$x = 4$

Now, for a square the apothem $({a_1})$ is half of the length of the side. Thus,
$({a_1}) = \dfrac{4}{2} = 2$
Now, calculating the length of the side of the equilateral triangle for given condition. We have,
Area = Perimeter (Given) … (2)
Let us assume that the length of the side is $y$.
For the equilateral triangle with sides $y$ area is $\dfrac{{\sqrt 3 }}{4}{y^2}$ and perimeter is $3y$.

From equation (2) we have,
$\dfrac{{\sqrt 3 }}{4}{y^2} = 3y$
Dividing both sides by $y$, we have,
$\dfrac{{\sqrt 3 }}{4}y = 3$
Dividing both sides by $\dfrac{{\sqrt 3 }}{4}$, we have,
$y = \dfrac{{12}}{{\sqrt 3 }}$ … (3)

Now, for an equilateral triangle the apothem $({a_2})$ is give as follows:
${a_2} = \dfrac{{\sqrt 3 }}{6}y$
Substituting the value from equation (3), we have,
${a_2} = \dfrac{{\sqrt 3 }}{6} \left(\dfrac{{12}}{{\sqrt 3 }}\right)$
${a_2} = 2$
Therefore, the apothems of both the equilateral triangle and square are equal.

Hence, the correct answer is option (A).

Note: While calculating the formula for apothem for various polygons it should be noted that the line from the centre falls perpendicular on the side of the polygon. The apothem of the square is simply half its side length.