### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #2 : How To Find The Length Of The Side Of A Hexagon

A regular hexagon has the same perimeter as the above right triangle. What is the length of one side of the hexagon?

**Possible Answers:**

The length cannot be determined from the information given.

**Correct answer:**

By the Pythagorean Theorem, the hypotenuse of the right triangle is

inches, making its perimeter

inches.

The regular hexagon, which has six sides of equal length, has the same perimeter, so each side measures

inches.

### Example Question #1 : Other Polygons

The length of a side of a regular octagon is one and a half times the hypotenuse of the above right triangle. Give the perimeter of the octagon in feet.

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem, the hypotenuse of the right triangle is

inches.

The sidelength of the octagon is therefore

inches,

and the perimeter of the regular octagon, which has eight sides of equal length, is

inches,

or

feet.

### Example Question #2 : Other Polygons

An equilateral triangle, a square, a regular pentagon, a regular hexagon, and a regular octagon have the same sidelength. Which is the greater quantity?

(A) The median of their perimeters

(B) The midrange of their perimeters

**Possible Answers:**

(A) is greater

(A) and (B) are equal

(B) is greater

It is impossible to tell which is greater from the information given

**Correct answer:**

(B) is greater

The answer is independent of the sidelength, so we can assume without loss of generality that the sidelength is 1. The equilateral triangle, the square, the pentagon, the hexagon, and the octagon have 3, 4, 5, 6, and 8 sides of equal length, respectively, so their perimeters are 3, 4, 5, 6, and 8.

The median of these perimeters is the middle perimeter, 5. The midrange of these perimeters is the mean of the greatest and the least perimeters:

The midrange, (B), is greater.

### Example Question #103 : Plane Geometry

A square, a regular pentagon, a regular hexagon, and a regular octagon have the same sidelength. Which is the greater quantity?

(A) The mean of their perimeters

(B) The median of their perimeters

**Possible Answers:**

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to tell which is greater from the information given

**Correct answer:**

(A) is greater

The answer is independent of the sidelength, so we can assume without loss of generality that the sidelength is 1. The square, the pentagon, the hexagon, and the octagon have 4, 5, 6, and 8 sides of equal length, respectively, so their perimeters are 4, 5, 6, and 8. The mean of these four perimeters is

units.

The median is the mean of the middle two perimeters, which are 5 and 6:

The mean, (A), is greater.

### Example Question #1 : Other Polygons

is a side of regular Pentagon as well as Square , which is completely outside Pentagon . is a side of equilateral , where is a point outside Square . Which is the greater quantity?

(a) The perimeter of Pentagon

(b) The perimeter of Pentagon

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

**Correct answer:**

(a) and (b) are equal

The figure referenced is below:

Pentagon is regular, so all of its sides have the same length; we will examine in particular. The perimeter of Pentagon is the sum of the lengths of its sides, which is .

Since is also a side of Square , it follows that ; since is also a side of equilateral , . The perimeter of Pentagon is equal to

,

the same as that of Pentagon .

### Example Question #4 : Other Polygons

A regular decagon has the same perimeter as the above right triangle. Give the length of one side.

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem, the hypotenuse of the right triangle is

inches, making its perimeter

inches.

A regular decagon has ten sides of equal length, so each side measures

inches.

### Example Question #101 : Geometry

A regular octagon has perimeter one meter. Which is the greater quantity?

(A) The length of one side

(B) 125 millimeters

**Possible Answers:**

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

**Correct answer:**

(A) and (B) are equal

A regular octagon has eight sides of equal length. The perimeter of this octagon is one meter, which is equal to 1,000 millimeters; each side, therefore, has length

millimeters

making the quantities equal.

### Example Question #105 : Plane Geometry

A regular pentagon has sidelength 72; the perimeter of a regular hexagon is 80% of that of the pentagon. Which is the greater quantity?

(A) The length of one side of the hexagon

(B) 50

**Possible Answers:**

(A) is greater

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

**Correct answer:**

(B) is greater

A regular pentagon has five sides of equal length; since one side is 72 units long, its perimeter is

.

80% of this is

,

so this is the length of the hexagon, and, since all six sides are of equal length, one side measures

(B) is greater.

### Example Question #7 : Other Polygons

A regular octagon has twice the perimeter of a regular pentagon. What is the ratio of the sidelength of the octagon to that of the pentagon?

**Possible Answers:**

**Correct answer:**

The solution is independent of the actual lengths, so we assume the pentagon has sidelength 1. Its perimeter is therefore 5. Subsequently, the octagon's perimeter is twice this, or 10, and its sidelength is one-eighth of this, or

.

The ratio of the sidelength of the octagon to that of the pentagon is

or 5 to 4.

### Example Question #106 : Plane Geometry

A regular octagon has perimeter one mile.Which is the greater quantity?

(a) The length of one side

(b) 880 feet

**Possible Answers:**

(a) and (b) are equal

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(b) is the greater quantity

**Correct answer:**

(b) is the greater quantity

A regular octagon has eight sides of equal length. The perimeter, which is the sum of the lengths of these sides, is one mile, which is equal to 5,280 feet. Therefore, the length of one side is

. This makes the length of a side less than 880 feet.