BTGmoderatorLU wrote:The perimeter of a polygon is 16. If the sides of the polygon are all of integer length, the shortest side of the polygon is 2 and the longest side of the polygon is 5, then the number of sides of the polygon could any number from?

A. 3 to 6
B. 4 to 5
C. 3 to 7
D. 4 to 6
E. 4 to 7

The OA is D.

Please, can someone explain this PS question? I'm confused about how can I solve it. Thanks in advance!

Given: The perimeter of a polygon is 16. The sides of the polygon are all of integer length, the shortest side of the polygon is 2 and the longest side of the polygon is 5.

To find out: Possible number of sides of the polygon.

Say the polygon has n sides; thus, we have to find out the minimum and the maximum value of n.

Since the perimeter is 16, the shortest side of the polygon is 2, and the longest side of the polygon is 5, the sum of the remaining (n - 2) sides = 16 - 2 - 5 = 9.

Case 1: Finding the minimum value of n

To find the minimum value of n, we must assume that the remaining (n - 2) sides are of 5 units (maximum possible dimension) each.

=> 9 = 5*(n - 2) => n = 3.8

Since n is an integer, the minimum value of n is 4.

Case 2: Finding the maximum value of n

To find the maximum value of n, we must assume that the remaining (n - 2) sides are of 2 units (minimum possible dimension) each.

=> 9 = 2*(n - 2) => n = 6.5

Since n is an integer, the maximum value of n is 6.

The correct answer: D

Hope this helps!

-Jay
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BTGmoderatorLU wrote:The perimeter of a polygon is 16. If the sides of the polygon are all of integer length, the shortest side of the polygon is 2 and the longest side of the polygon is 5, then the number of sides of the polygon could any number from?

A. 3 to 6
B. 4 to 5
C. 3 to 7
D. 4 to 6
E. 4 to 7

The OA is D.

Please, can someone explain this PS question? I'm confused about how can I solve it. Thanks in advance!

Since a polygon can have as few as 3 sides, let's start with a triangle. If two sides of the triangle are 2 and 5, then the third side would have to be 16 - 2- 5 = 9. However, this won't work because the longest side of the polygon is supposed to be 5.

Let's consider a 4-sided figure. To find the minimum number of sides, let's make the length of the sides as large as possible. We see that if three sides have a length of 5, the polygon will have a perimeter of 5 + 5 + 5 + 2 = 17, which is greater than (but very close to) the given perimeter. If two sides have a length of 5, one side has a length of 4, and one side has a length of 2, then the polygon has a perimeter of 5 + 5 + 4 + 2 = 16. So 4 is the minimum number of sides the polygon can have.

To find the maximum number of sides, let's make the lengths of the sides as small as possible. We see that if five sides have the minimum length, which is 2, then the perimeter is 2 + 2 + 2 + 2 + 2 + 5 = 15, which is less than (but again very close to) the given perimeter. If four sides have a length of 2, one side has a length of 3 and one side has the maximum length, then the perimeter is 2 + 2 + 2 + 2 + 3 + 5 = 16. Thus, the maximum number of sides this polygon can have is 6.

Answer: D