Quadrilateral ABCD has a perimeter of 26 and sides of integer lengths. If AB=m and BC=CD=DA=n, then what is the difference between the greatest and least possible values of n?
A) 7
B) 6
C) 5
D) 4
E) 3
The perimeter is: 3n + m = 26
So: n = (26-m)/3
Thus n must be an integer value such that:
1/3 < n < 25/3
Find the least and greatest integers in the domain
Well, let's start by finding the maximum possible value of n. Since the perimeter of ABCD is 26, we know that m + n + n + n = 26. Simplifying this equation gives us m + 3n = 26.
In order to maximize n, we want to minimize m. Since both m and n have to be integers, the smallest possible value for m is 1 (because any smaller and we wouldn't have enough room for n to be an integer).
So, if m = 1, then we have 1 + 3n = 26. Solving this equation gives us n = 8.
Now, let's find the minimum possible value of n. Again, we have m + 3n = 26. But this time, we want to maximize m and minimize n. The largest possible value for m is 12 (because any larger and we wouldn't have enough room for n to be an integer).
So, if m = 12, then we have 12 + 3n = 26. Solving this equation gives us n = 4.
Therefore, the difference between the greatest and least possible values of n is 8 - 4 = 4. So the answer is D) 4.
But hey, look at the bright side - no matter what, the difference between the greatest and least values of n is less than the number of fingers on one hand. So, it's not that big of a difference after all!
To find the difference between the greatest and least possible values of n, we need to consider the possible lengths of sides AB, BC, CD, and DA.
Since the perimeter of the quadrilateral is 26, we have the equation:
AB + BC + CD + DA = 26
Let's assume that AB = m and BC = CD = DA = n.
Substituting these values into the equation, we get:
m + n + n + n = 26
m + 3n = 26
To find the greatest and least possible values of n, we need to find the minimum and maximum values of n that satisfy this equation.
Since the sides must have integer lengths, n must be an integer. Therefore, the minimum possible value of n is 1.
Substituting n = 1 into the equation, we have:
m + 3(1) = 26
m + 3 = 26
m = 26 - 3
m = 23
Therefore, the possible lengths of the sides are AB = 23 and BC = CD = DA = 1, resulting in the minimum value of n.
To find the maximum possible value of n, we can assume that m = 1. Substituting m = 1 into the equation, we have:
1 + 3n = 26
3n = 26 - 1
3n = 25
n = 25/3
Since n must be an integer, the greatest possible value of n is 8 (rounded down from 25/3).
Therefore, the difference between the greatest and least possible values of n is 8 - 1 = 7.
The correct answer is A) 7.
To solve this question, we need to apply the properties of a quadrilateral and the given information.
The perimeter of a quadrilateral is the sum of the lengths of all its sides. In this case, the perimeter of ABCD is given as 26.
Let's set up the equation using the given information: AB + BC + CD + DA = 26.
Since AB = m and BC = CD = DA = n, we can rewrite the equation as m + 3n = 26.
To find the greatest and least possible values of n, we should try to maximize and minimize the value of n while satisfying the equation.
To maximize n, we should minimize m. The question doesn't provide any restrictions on m, so we can set it as small as possible, which is m = 1.
Now we have the equation 1 + 3n = 26. Solving this equation, we find n = 25/3. Since we are dealing with integer lengths, we can consider n as the highest possible value by rounding up to the next integer, which is n = 9.
To minimize n, we should maximize m. Since m and n are both positive integers, we can set m as high as possible, which is m = 12.
Using the equation 12 + 3n = 26, we find n = 14/3. Again, we round down to the previous integer because we are dealing with integer lengths. Therefore, n is minimum when n = 4.
Now we can find the difference between the greatest and least possible values of n:
Difference = Maximum value of n - Minimum value of n
Difference = 9 - 4
Difference = 5
Therefore, the difference between the greatest and least possible values of n is 5.
The correct answer is C) 5.