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. $4/3$ times the second
• C. $\frac{\sqrt{2}}{\sqrt{3}}$ times the second
• D. $\frac{2}{\sqrt{3}}$ times the second
• E. indeterminately related to the second

Answer 1

The correct option is A equal to the second

Let $s_1$ be the side of the square, $a_1$ its apothem; then $s_1^2=4s_1$ and since $2a_1=s_1$, we have $4a_1^2=8a_1$ or $a_1=2$. Let $s_2$ be the side of the equilateral triangle, h its altitude and $a_2$ its apothem; then $\frac{s_2^2\sqrt{3}}{4}=3s_2$. Since $h=3a_2$, and $s_2=\frac{2h}{\sqrt{3}}=\frac{6a_2}{\sqrt{3}}$, we have $\left({36}_2^2/3\right)\sqrt{3}/4=3.6\Rightarrow a_2/\sqrt{3}and{a}_2=2=a_1$