Refer to the figure below, in which \(\displaystyle \overline{EB}\) has been constructed, and the top and right sides of the figure have been extended to their intersection \(\displaystyle F\) to form Rectangle \(\displaystyle AFDE\).

Assume Statement 1 alone. Since opposite sides of a rectangle have the same length, \(\displaystyle FD =EA= 12\). By segment addition, \(\displaystyle FC+CD= FD\), and, since \(\displaystyle CD = 8\), by substitution, \(\displaystyle FC+8= 12\). Therefore, \(\displaystyle FC = 4\), and the Pythagorean Theorem can be used to find \(\displaystyle BF\):

\(\displaystyle BF = \sqrt{(BC)^{2} - (FC)^{2}} = \sqrt{5^{2} - 4^{2}} = \sqrt{25-16} = \sqrt{9} = 3\)

Since opposite sides of a rectangle have the same length, \(\displaystyle AF = ED = 12\), and by segment addition, \(\displaystyle AB + BF = AF\). By substitution, \(\displaystyle AB + 3= 12\), and \(\displaystyle AB = 9\).

Assume Statement 2 alone. Since \(\displaystyle \overline{EB}\), the hypotenuse of right triangle \(\displaystyle \bigtriangleup ABE\), and \(\displaystyle \overline{A E}\), one of its legs, have lengths 15 and 12, respectively, the length of the other leg \(\displaystyle \overline{A B}\) can be found using the Pythagorean Theorem:

\(\displaystyle AB = \sqrt{(EB)^{2} - (AE)^{2}} = \sqrt{15^{2} - 12^{2}} = \sqrt{225-144} = \sqrt{81} = 9\)

We can construct perpendicular line segments from \(\displaystyle B\) to \(\displaystyle \overline{ED}\) and from \(\displaystyle C\) to \(\displaystyle \overline{AE}\) as follows:

\(\displaystyle \overline{BC}\) is the hypotenuse of a right triangle \(\displaystyle \bigtriangleup BGC\), so if we can determine the lengths of \(\displaystyle \overline{BG}\) and \(\displaystyle \overline{GC}\), we can use the Pythagorean Theorem to determine the length of \(\displaystyle \overline{BC}\).

Assume Statement 1 alone. By segment addition, \(\displaystyle EF+FD = ED = 12\).  Since \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{EF}\) are opposite sides of a rectangle, \(\displaystyle AB = EF\); similarly, \(\displaystyle GC = FD\). It follows by substitution that  \(\displaystyle AB+GC = 12\). Since \(\displaystyle AB = 7\), it follows that \(\displaystyle 7+GC = 12\), and \(\displaystyle GC = 5\). However, no additional information exists to find \(\displaystyle BG\).

Assume Statement alone. By similar reasoning, \(\displaystyle BG + CD = 20\); since \(\displaystyle CD = 12\), \(\displaystyle BG + 12= 20\), and \(\displaystyle BG = 8\). However, no information exists to find \(\displaystyle GC\).

The two statements put together, however, yield both necessary values: \(\displaystyle BG = 8\) and \(\displaystyle GC = 5\). By the Pythagorean Theorem,

\(\displaystyle BC = \sqrt{(BG)^{2}+(GC)^{2}} = \sqrt{8^{2}+5^{2}} = \sqrt{64+25} = \sqrt{89}\).

To find the length of \(\displaystyle \overline{CD}\), we can extend \(\displaystyle \overline{BC}\) to meet \(\displaystyle \overline{EF}\) at a point \(\displaystyle G\) to form two rectangles, as seen below:

Statement 1 gives no helpful information, since the length of \(\displaystyle \overline{DE}\), which is not parallel to \(\displaystyle \overline{CD}\), has no bearing on that side's length.

If we are given Statement 2 alone, then, as seen in the diagram, \(\displaystyle FG + GE = FE = 12\) from segment addition, \(\displaystyle AB = 7\), from Statement 2, and from congruence of opposite sides of a rectangle, \(\displaystyle AB = FG\) and \(\displaystyle CD = GE\). Therefore, \(\displaystyle AB + CD = 12\), \(\displaystyle 7 + CD = 12\), and \(\displaystyle CD = 5\). 

Below is a regular hexagon \(\displaystyle HEXAGN\), with its three diameters, its center \(\displaystyle O\), and its circumscribed circle, which also has center \(\displaystyle O\).

If Statement 1 is true. then the circle, with circumference \(\displaystyle 12 \pi\), has as its diameter \(\displaystyle 12 \pi \div \pi = 12\), which is 12; this makes the two statements equivalent, so we need only establish that one statement is sufficient or insufficient.

Either way, \(\displaystyle OA\), the radius of the hexagon, is 6. The six triangles that are formed by the sides and diameters of a regular hexagon are all equilateral by symmetry, so each side of the hexagon - in particular, \(\displaystyle \overline{HE}\) - has length 6.

Assume both statements to be true, and examine these two scenarios:

Case 1: The hexagon could have six sides of length 7.

Case 2: The hexagon has four sides of length 7, one of which is \(\displaystyle \overline{EX}\), one side of length 6, and one side - \(\displaystyle \overline{HE}\) - of length 8. 

In both situations, \(\displaystyle EX= 7\) and the perimeter of the hexagon is 42:

\(\displaystyle 7 \times 6= 7 \times 4+6+8 = 42\).

The conditions of both statements would be met in both scenarios, so the two statements together are insufficient.

Statement 1 alone states that \(\displaystyle \overline{PE}\) is congruent to two other sides, but gives no actual measurements. Statement 2 alone gives the actual measurements of two other segments, but without further information, such as how their lengths relates to that of \(\displaystyle \overline{PE}\), no information about \(\displaystyle \overline{PE}\) can be inferred.

Now, assume both statements to be true. From Statement 2, \(\displaystyle \overline{TN}\) has length 10, and from Statement 2, \(\displaystyle \overline{NT}\), which is the same line segment (which can be named after its endpoints in either order), has the same length as \(\displaystyle \overline{PE}\). Therefore, \(\displaystyle PE = 10\).

Statement 1 alone only gives the perimeter - the sum of the lengths of the sides - but gives no information about the individual sidelengths. (In particular, there is no indication that the pentagon is regular).

Assume Statement 2 alone. \(\displaystyle \overline{PE}\) and \(\displaystyle \overline{EP}\) are two names for the same line segment, which can be named after its endpoints in either order. Therefore, \(\displaystyle PE = EP = 10\).

Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with \(\displaystyle \overline{NT}\) the side of length 12, then \(\displaystyle \overline{NT}\) is the shortest side, and the perimeter is 

\(\displaystyle 12+13+13+13+14 = 65\).

On the other hand, if the pentagon has sides of length 11, 12, 13, 14, and 15, 

with \(\displaystyle \overline{NT}\) the side of length 12, then \(\displaystyle \overline{NT}\) is not the shortest side, and the perimeter is 

\(\displaystyle 11+ 12+13+14 +15= 65\).

The two statements together are insufficient.

Statement 1 alone only gives information about one side of the hexagon, and Statement 2 gives only information about the perimeter without giving any clues as to the individual sidelengths; neither is sufficient to answer the question.

Assume both statements are true. If \(\displaystyle \overline{XA}\), with length 10, is the longest side of Hexagon \(\displaystyle HEXAGO\), then 

\(\displaystyle HE, EX, AG, GO, OH < 10\)

By the addition property of inequality, 

\(\displaystyle HE+ EX+XA+ AG+ GO+ OH < 10 +10 +10 +10 +10 +10 = 60 < 66\)

This means the sum of the sidelengths of the hexagon, which is its perimeter, is less than 66, in contradiction to Statement 2. \(\displaystyle \overline{XA}\) cannot be the longest side.