How to find the length of a side - ISEE Upper Level (grades 9-12) Quantitative Reasoning
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A regular decagon has the same perimeter as the above right triangle. Give the length of one side.
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.
By the Pythagorean Theorem, the hypotenuse of the right triangle is
A regular decagon has ten sides of equal length, so each side measures
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A regular octagon has perimeter one meter. Which is the greater quantity?
(A) The length of one side
(B) 125 millimeters
A regular octagon has perimeter one meter. Which is the greater quantity?
(A) The length of one side
(B) 125 millimeters
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.
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
making the quantities equal.
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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 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 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.
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.
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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?
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.
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
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A regular octagon has perimeter one mile.Which is the greater quantity?
(a) The length of one side
(b) 880 feet
A regular octagon has perimeter one mile.Which is the greater quantity?
(a) The length of one side
(b) 880 feet
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.
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
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