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The sides of a triangle have lengths 1, 2 and x. If x is an integer

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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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New post 01 Sep 2017, 00:42
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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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Triangle inequality theorem states that
the sum of any 2 sides of a triangle must be greater than the measure of the third side!
If the three sides are x,y, and z, then
x+y>z
x+z>y
y+z>x

So if x+1>2 and x+2>1, x has to be greater than 1
But, 2+1 is also greater than x.
Range of x is 1 < x < 3
The only possible value of x is 2(Option 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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New post 01 Sep 2017, 06:51
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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New post 01 Sep 2017, 14:41
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

Last edited by genxer123 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]

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New post 01 Sep 2017, 14:52
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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


IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
|A - B| < third side < A + B

So, we can write: |2 - 1| < x < 2 + 1
Simplify: 3 < x < 1

Since x must be an integer, x must equal 2

Answer:
[Reveal] Spoiler:
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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New post 05 Sep 2017, 17:41
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]

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New post 05 Sep 2017, 18:58
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]

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New post 07 Sep 2017, 13:44
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] 07 Sep 2017, 13:44
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