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?
The length cannot be determined from the information given.
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.
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
(A) is greater
(A) and (B) are equal
(B) is greater
It is impossible to tell which is greater from the information given
(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
(A) is greater
(B) is greater
(A) and (B) are equal
It is impossible to tell which is greater from the information given
(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
(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
(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.
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
It is impossible to determine which is greater from the information given
(B) is greater
(A) and (B) are equal
(A) is greater
(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
(A) is greater
It is impossible to determine which is greater from the information given
(B) is greater
(A) and (B) are equal
(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?
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
(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
(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.