Each of the two statements allows you to find out the perimeter of one of the polygons by multiplying its sidelength by the number of its sides. However, neither statement offers any clues to the perimeter of the other polygon. Both statements together, however, allow you to determine and to compare both perimeters.

The area of a polygon is one-half the product of its perimeter and its apothem (the perpendicular distance from the center to a side). Therefore,

\(\displaystyle A = \frac{1}{2}ap\), 

or

\(\displaystyle \frac{1}{2}ap =A\)

\(\displaystyle \frac{1}{2}ap \cdot \frac{2}{a} =A\cdot \frac{2}{a}\)

\(\displaystyle p = \frac{2A}{a}\)

Therefore, the perimeter can be determined from both area and apothem, but not from one alone. Neither statement alone gives you enough information, but from both statements together, it can be determined that the pentagon has the greater perimeter.

The figure can be seen as a \(\displaystyle d \times e\) rectangle cut out of an \(\displaystyle a \times b\) rectangle.

The perimeter of the composite figure is

\(\displaystyle P = a + b + c + d + e + f\).

However, since opposite sides of a rectangle are congruent, then, as can be seen in the figure, \(\displaystyle c + e = b\) and \(\displaystyle d+f = a\).

The perimeter can then be rewritten:

\(\displaystyle P = a + b + c + d + e + f\)

\(\displaystyle = a + d + f + b + c + e\)

\(\displaystyle = a + a + b + b\)

\(\displaystyle =2a + 2b\)

Therefore, it is necessary and sufficient to know \(\displaystyle a\) and \(\displaystyle b\); the other four sidelengths are not needed to determine the perimeter of the figure.

To find perimeter, we need the length of all the sides. Note that we are dealing with a regular polygon, so all of its sides are equal.

Statement I gives us one side length. Multiply the side by ten (decagons have ten sides):

\(\displaystyle p=56\cdot 10=560\)

Statement II gives us the length of two sides together. Multiply by 5 to get the total perimeter:

\(\displaystyle p=112\cdot 5=560\)

Consider pentagon \(\displaystyle SPLIT\).

I) Side \(\displaystyle SP\) has the same length as side \(\displaystyle TS\), 5 inches.

II) Side \(\displaystyle TS\) is one-third the length of sides \(\displaystyle PL\), \(\displaystyle LI\) and \(\displaystyle IT\).

What is the perimeter of \(\displaystyle SPLIT\)? 

To find perimeter, we need to know the length of all the sides.

Statement I gives us the lengths of two sides.

Statement II relates one of the sides given in Statement I to the other three sides:

\(\displaystyle 3TS=PL=LI=LT=3\cdot 5=15\)

\(\displaystyle P=5+5+15+15+15=55in\)

The answer can be seen more easily by constructing the altitude of \(\displaystyle \bigtriangleup ABC\) from \(\displaystyle B\), as seen below:

Each interior angle of a hexagon measures \(\displaystyle 120^{\circ }\),and the altitude also bisects \(\displaystyle \angle ABC\), the vertex angle of isosceles \(\displaystyle \bigtriangleup ABC\). \(\displaystyle \bigtriangleup ABM\) is easily proved to be a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem,

\(\displaystyle BM = \frac{1}{2} \cdot AB = \frac{1}{2} \cdot 10 = 5\)

\(\displaystyle AM = BM \cdot \sqrt{3} = 5\sqrt{3}\)

The altitude also bisects \(\displaystyle \overline{AC}\) at \(\displaystyle M\), so

\(\displaystyle AC = 2 \cdot AM = 2 \cdot 5 \sqrt{3} = 10 \sqrt{3}\).

Statement 1:  The perimeter is 10.

Given the perimeter of a rectangle is 10, set up the equation such that the sum of twice of length and width are equal to 10.

\(\displaystyle 2L+2W=10\)

There are multiple combinations of length and width that will work in this scenario.

Statement 2:  The area is 4.

Write the formula for area of a rectangle and substitute the area.

\(\displaystyle A=LW\)

\(\displaystyle 4=LW\)

Statements 1 and 2 require each other in order to solve for the length and width since there are two unknown variables.

Afterward, the Pythagorean Theorem can be used to solve for the diagonal of the rectangle.

Therefore:

\(\displaystyle \textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}\)

If the pentagon is regular, then it is known that each of the internal angles is \(\displaystyle 108\) degrees, since the sum of the five interior angles of a pentagon is \(\displaystyle 540\) degrees. Statement 2 does not give new information, nor does it give enough.

Statement 1 does, however. The length of the diagonal can be found by using the law of cosines:

\(\displaystyle c^2=a^2+b^2-2abcos(C)\)

\(\displaystyle a=5, b=5, C=108\)

and c is the length of the diagonal.

\(\displaystyle d=\sqrt{5^2+5^2-2(5)(5)cos(108)}\)

The relationship between the number of sides of a regular polygon \(\displaystyle n\) and the measure of a single angle \(\displaystyle A\) is

 \(\displaystyle A = \frac{180(n-2)}{n}\) 

If we are given that \(\displaystyle A = 140\), then we can substitute and solve for  \(\displaystyle n\) :

\(\displaystyle 140 = \frac{180(n-2)}{n}\)

\(\displaystyle 140 = \frac{180n-360}{n}\)

\(\displaystyle 140n = 180n-360\)

\(\displaystyle (140-180)n = -360\)

\(\displaystyle -40n = -360\)

\(\displaystyle n = 9\)

making the figure a nine-sided polygon.

Knowing only the measure of each side is neither necessary nor helpful; for example, it is possible to construct an equilateral triangle with sidelength 8 or a square with sidelength 8. 

The correct answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2 alone.

Information about the hexagon is irrelevant, so Statement 2 has no bearing on the answer to the question.

The measures of the exterior angles of any pentagon, one per vertex, total \(\displaystyle 360^{\circ }\), and they are congruent if the pentagon is regular, so if this is the case, each would measure \(\displaystyle 360 \div 5 = 72^{\circ }\). But if Statement 1 is true, then an exterior angle of the pentagon measures \(\displaystyle 70^{\circ}\) . Therefore, Statement 1 is enough to answer the question in the negative.