1951 AMC 12 Problem 25

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:

Equal to the second

\dfrac43

times the second

\dfrac{2}{\sqrt3}

times the second

\dfrac{\sqrt2}{\sqrt3}

times the second

Indeterminately related to the second

Answer: A

Concepts:regular polygon area perimeter

Difficulty rating: 1610

Small hint:

For a regular polygon, area equals one half the apothem times the perimeter

Big hint:

If a nonzero perimeter equals the area, cancel the perimeter from

A=\frac12rP

Solution:

Every regular polygon with apothem

r

and perimeter

P

has area

A=\frac12rP.

If its numerical area equals its nonzero perimeter, then

\frac12rP=P,

so

r=2.

This applies to both the square and the equilateral triangle, so their apothems are equal.

Thus, the correct answer is A.

1951 AMC 12 Problem 25

Answer: A

Difficulty rating: 1610

Small hint:

Big hint:

Solution:

Problem 25 in other years