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The sides of a triangle have lengths 1, 2 and x. If x is an integer : Problem Solving (PS)
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
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Bunuel wrote:
The sides of a triangle have lengths 1, 2 and x. If x is an integer, what is the value of x?

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1


Property: Sum of two shorter sides > Third biggest side

i.e. x+1 > 2

i.e. x > 1

But also 1+2 > x [If x is the longest dimension]

i.e. 3 > x

i.e. 1 < x < 3

Hence, x must be 2

Answer: Option D
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The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
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Bunuel wrote:
The sides of a triangle have lengths 1, 2 and x. If x is an integer, what is the value of x?

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1

There is a quick way to set up the possibilities for side length x, given the triangle inequality theorem, which will be in form

? < x < ?

Subtract one known side from the other known side. The answer is the first (leftmost) term of the compound inequality. (2 - 1) = 1

Then add the two known sides. The answer is the third term (rightmost) of the inequality. (1 + 2) = 3

Done.

1 < x < 3
x = 2

Answer D

Originally posted by generis on 01 Sep 2017, 14:41.
Last edited by generis on 01 Sep 2017, 15:06, edited 1 time in total.
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
Expert Reply
Bunuel wrote:
The sides of a triangle have lengths 1, 2 and x. If x is an integer, what is the value of x?

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1


Since the sum of any two sides of a triangle must be greater than the third side, the only integer value of x that works is 2.

Answer: D
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
In triangle sum of two sides > one side > difference of two sides

=> 1+2>x>2-1
=> 3>x>1

X is an integer => x = 2
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
One of the property of triangle : Sum of two sides must be greater than 3rd side.

As we have 1 and 2 as two sides of the triangle. Now the third side would be 1+2 > third side ( 1 or 2).

If we take combination as 1 2 1 , there is contradiction as 1+1 = 2 ( sum of two sides must be greater than third side).

So the third side would be 2.

correct option : D
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
The triangle property says difference of 2 sides is smaller than the 3rd side therefore 2-1 =1 so it means that the 3rd side is greater than 1.
Also the property of the triangle says the sum of 2 sides is greater than the 3rd side. Therfore 2+ 1= 3 is greater than the 3rd side.
SO basically it means the 3rd side cant be 1 and it cant be 3 . The only integer between 3 and 1 =2 . Therfore d is the answer = 2
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
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Range of x will be : 2 - 1 < x < 2 + 1

=> 1 < x < 3

Hence, x = 2

Answer D
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
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Re: The sides of a triangle have lengths 1, 2 and x. If x is an integer [#permalink]
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