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605-655 (Medium)|   Geometry|      
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nick1816
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In a triangle with integer side lengths, one side is three times as Updated on: 26 Apr 2019, 07:09
Originally posted by nick1816 on 26 Apr 2019, 06:32.
Last edited by nick1816 on 26 Apr 2019, 07:09, edited 1 time in total.
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In a triangle with integer side lengths, one side is three times as long as a second side, and the length of third side is 17. What is the difference between the greatest and smallest possible perimeter of the triangle?

A. 15
B. 12
C. 25
D. 15
E. 13
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B
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In a triangle with integer side lengths, one side is three times as 26 Apr 2019, 06:54
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nick1816
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of third side is 17. what is the greatest possible perimeter of the triangle?

A. 48
B. 49
C. 52
D. 55
E. 50
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Almost the same question: https://gmatclub.com/forum/in-a-triangl ... 91755.html
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In a triangle with integer side lengths, one side is three times as 26 Apr 2019, 07:31
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let the other two side be x and 3x

then x+3x+17 = P ( perimeter )

now value is minimum when x = 1

Minimum P = 21

Now 4x+17 = 21

Now for the maximum valyue or perimeter , we should add the difference to minimum value and see if we can find the integer value for x

i.e

4x= 21 + ( value from options)-17

we can see that we get integer value only at x = 12


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In a triangle with integer side lengths, one side is three times as 29 Aug 2020, 05:25
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In a triangle with integer side lengths, one side is three times as long as a second side, and the length of third side is 17. What is the difference between the greatest and smallest possible perimeter of the triangle?

A. 15
B. 12
C. 25
D. 15
E. 13[/quote]

Three sides of a triangle are a,b,and c.
First side is a
Second side is b
Third side is c
a = 3b, b= b,c=17 Given

a -b < c < a+b by side property from triangle
2b < 17 < 4b
Perimeter of a triangle is a+b+c
so, maximum perimeter will be 17 + 4b and minimum perimeter will be 17 + 2b
Difference between maximum and minimum is (17 + 4b) - ( 17+2b) = 2b

so our answer will be multiple of 2.
Hence, option B
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In a triangle with integer side lengths, one side is three times as 29 Aug 2020, 06:26
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When we subtract the two perimeters, the '17' will subtract away. So we'll have our smallest triangle, with sides x and 3x, and our biggest triangle, with sides y and 3y, and the difference in perimeters will be 4y - 4x, which must be even. So only one answer is possible, 12.

To get that answer exactly: if we make x as small as possible, we need the sum of the two shortest sides to exceed the third side. So if our shortest sides are x and 3x, we need their sum, 4x, to be greater than 17, and x must be at least 5.

If we make x as large as possible, we need the sum of the two short sides, x and 17, to exceed the third side, 3x. So x + 17 > 3x must be true, and x can be at most 8 if x is an integer.

So the smallest possible triangle has lengths 5, 15, 17, and the largest has lengths 8, 17, 24, and their perimeters differ by 12.
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In a triangle with integer side lengths, one side is three times as 30 Jan 2022, 14:36
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