DSQ: Calculating the length of a side of a polygon - GMAT Quantitative

Below is a regular hexagon , with its three diameters, its center , and its circumscribed circle, which also has center .

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

Either way, , 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, - has length 6.

We can construct perpendicular line segments from to and from to as follows:

is the hypotenuse of a right triangle , so if we can determine the lengths of and , we can use the Pythagorean Theorem to determine the length of .

Assume Statement 1 alone. By segment addition, . Since and are opposite sides of a rectangle, ; similarly, . It follows by substitution that . Since , it follows that , and . However, no additional information exists to find .

Assume Statement alone. By similar reasoning, ; since , , and . However, no information exists to find .

The two statements put together, however, yield both necessary values: and . By the Pythagorean Theorem,

.

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 , one side of length 6, and one side - - of length 8.

In both situations, and the perimeter of the hexagon is 42:

.

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

Statement 1 alone states that 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 , no information about can be inferred.

Now, assume both statements to be true. From Statement 2, has length 10, and from Statement 2, , which is the same line segment (which can be named after its endpoints in either order), has the same length as . Therefore, .

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. and are two names for the same line segment, which can be named after its endpoints in either order. Therefore, .

Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with the side of length 12, then is the shortest side, and the perimeter is

.

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

with the side of length 12, then is not the shortest side, and the perimeter is

.

The two statements together are insufficient.

Refer to the figure below, in which has been constructed, and the top and right sides of the figure have been extended to their intersection to form Rectangle .

Assume Statement 1 alone. Since opposite sides of a rectangle have the same length, . By segment addition, , and, since , by substitution, . Therefore, , and the Pythagorean Theorem can be used to find :

Since opposite sides of a rectangle have the same length, , and by segment addition, . By substitution, , and .

Assume Statement 2 alone. Since , the hypotenuse of right triangle , and , one of its legs, have lengths 15 and 12, respectively, the length of the other leg can be found using the Pythagorean Theorem:

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 , with length 10, is the longest side of Hexagon , then

By the addition property of inequality,

This means the sum of the sidelengths of the hexagon, which is its perimeter, is less than 66, in contradiction to Statement 2. cannot be the longest side.

To find the length of , we can extend to meet at a point to form two rectangles, as seen below:

Statement 1 gives no helpful information, since the length of , which is not parallel to , has no bearing on that side's length.

If we are given Statement 2 alone, then, as seen in the diagram, from segment addition, , from Statement 2, and from congruence of opposite sides of a rectangle, and . Therefore, , , and .

Below is a regular hexagon , with its three diameters, its center , and its circumscribed circle, which also has center .

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

Either way, , 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, - has length 6.

We can construct perpendicular line segments from to and from to as follows:

is the hypotenuse of a right triangle , so if we can determine the lengths of and , we can use the Pythagorean Theorem to determine the length of .

Assume Statement 1 alone. By segment addition, . Since and are opposite sides of a rectangle, ; similarly, . It follows by substitution that . Since , it follows that , and . However, no additional information exists to find .

Assume Statement alone. By similar reasoning, ; since , , and . However, no information exists to find .

The two statements put together, however, yield both necessary values: and . By the Pythagorean Theorem,

.

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 , one side of length 6, and one side - - of length 8.

In both situations, and the perimeter of the hexagon is 42:

.

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

Statement 1 alone states that 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 , no information about can be inferred.

Now, assume both statements to be true. From Statement 2, has length 10, and from Statement 2, , which is the same line segment (which can be named after its endpoints in either order), has the same length as . Therefore, .

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. and are two names for the same line segment, which can be named after its endpoints in either order. Therefore, .

Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with the side of length 12, then is the shortest side, and the perimeter is

.

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

with the side of length 12, then is not the shortest side, and the perimeter is

.

The two statements together are insufficient.

Refer to the figure below, in which has been constructed, and the top and right sides of the figure have been extended to their intersection to form Rectangle .

Assume Statement 1 alone. Since opposite sides of a rectangle have the same length, . By segment addition, , and, since , by substitution, . Therefore, , and the Pythagorean Theorem can be used to find :

Since opposite sides of a rectangle have the same length, , and by segment addition, . By substitution, , and .

Assume Statement 2 alone. Since , the hypotenuse of right triangle , and , one of its legs, have lengths 15 and 12, respectively, the length of the other leg can be found using the Pythagorean Theorem:

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 , with length 10, is the longest side of Hexagon , then

By the addition property of inequality,

This means the sum of the sidelengths of the hexagon, which is its perimeter, is less than 66, in contradiction to Statement 2. cannot be the longest side.

Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with the side of length 12, then is the shortest side, and the perimeter is

.

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

with the side of length 12, then is not the shortest side, and the perimeter is

.

The two statements together are insufficient.

To find the length of , we can extend to meet at a point to form two rectangles, as seen below:

Statement 1 gives no helpful information, since the length of , which is not parallel to , has no bearing on that side's length.

If we are given Statement 2 alone, then, as seen in the diagram, from segment addition, , from Statement 2, and from congruence of opposite sides of a rectangle, and . Therefore, , , and .