Geometry Question

Question

A square and an equilateral triangle each have apothem of length 1cm. Which of the following statements is always true?
I: The perimeter of the square is greater than the perimeter of the triangle.
II: The area of the square is less than the area of the triangle.
III: The radius of the square is less than the radius of the triangle.
a. I only
b. II only
c. I and II only
d. II and III only

Answer 1

Whenever the word 'apothem' is used in a Geometry question, it usually is referring to a regular polygon inscribed within a circle where the vertices touch the circumference of the circle.  In this question we are comparing a square and an equilateral triangle where both have an apothem of 1 cm.  Our gut feeling may be that the square is larger.  However, in this case, we simply must do the math.

Joseph C. was correct in stating this question was worded very poorly and the key is "Are the polygons inscribed within the circle or is the circle inscribed within the polygons?"

However, I believe Joseph C. may have overlooked the fact that there are two different shapes in this question and his calculation for the side was only completed for the square (2 cm) and not the equilateral triangle.  If he had solved for the side of the triangle he would have found its sides were approximately 3.464 cm.

To find the perimeters:  For the square it is 4 times 2 cm = 8 cm.  For the triangle it is 3 times 3.464 = 10.392 cm.

Therefore answer a. is incorrect.

How about the areas?  The square area is base times height or 2 times 2 = 4 cm²

                                 The triangle is 1/2 the base times height or 1/2 3.464 times 3 = about 5.196 cm²

 

So answer b. is correct, but we must consider the radii of the square and triangle.

For the square:  The line drawn from the center of the of the square to the vertex forms an isosceles triangle with 45 degree angles and a 90 degree angle.  This is a special triangle and the hypotenuse is equal to the length of the leg times √2 or approximately 1.414 cm.  Therefore the radii of the circle the square is inscribed within is about 1.414 cm.

For the triangle: Measuring from the center of the triangle to one of its vertex points touching the circle it is inscribed within we find its radii is 2 cm.

Therefore answer d. ll and lll only   is the correct answer.

Again, this question was poorly written.  However, using the reasoning of both polygons being inscribed within circles and both having an apothem of 1 cm, answer d. is the only correct answer.

Answer 2

An apothem is a line segment from the center of a regular polygon to the midpoint of a side.
The apothem length is given by the formula:
apothem = s/[2(tan 180/n)]
where:
s is the length of any side
n is the number of sides
tan is the tangent function calculated in degrees

 

We can use that formula to solve for s, since we know the apothem, and s = 2cm

 

We now know the side length of the two figures

1. yes, the perimeter of the square is greater than the perimeter of the triangle

2. no, the area of the square is NOT less than the area of the triangle

3. "Radius" is a term most commonly associated with circles. When used in reference to a square, the radius is the distance from the center of the square to any of the four vertices, or where two sides intersect. In the case of a triangle, the radius is the radius of an inscribed circle, which is the apothem in this case. so the radius of the square will be longer than its apothem.

 

So, the answer to the question is option a