Problem: 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:
Answer Choices:
A. equal to the second
B. 34​ times the second
C. 3​2​ times the second
D. 3​2​​ times the second
E. indeterminately related to the second
Solution:
Let s1​ be the side of the square, a1​ its apothem; then s12​=4s1​ and since 2a1​=s1​, we have 4a12​=8a1​ or a1​=2. Let s2​ be the side of the equilateral triangle, h its altitude and a2​ its apothem; then s22​3​/4=3s2​. Since h=3a2​, and s2​=2h/3​=6a2​/3​, we have (36a22​/3)3​/4=3⋅6a2​/3​, and a2​=2=a1​.