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# If AB=3, what is the shortest side of triangle ABC?

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If AB=3, what is the shortest side of triangle ABC? [#permalink]  18 Aug 2013, 11:05
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If AB=3, what is the shortest side of triangle ABC?

(1) AC=5

(2) The perimeter of ΔABC is 13.
[Reveal] Spoiler: OA

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Re: If AB=3, what is the shortest side of triangle ABC? (1) AC= [#permalink]  18 Aug 2013, 11:21
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blueseas wrote:
If AB=3, what is the shortest side of triangle ABC?

(1) AC=5

(2) The perimeter of ΔABC is 13.

From F.S 1, we get a valid triangle for AC=5 and BC=4, and the smallest side is AB.
Again, for BC=2.1, we get another valid triangle and this time, the smallest side is BC. As we get 2 different answers, Insufficient.

From F.S 2, we know that AC+BC = 13-3 = 10. Now, we know that the difference of 2 sides must be less than the third side of a valid triangle.

Thus, as have |AC-BC|<3. Replacing the value of AC, we get $|10-BC-BC|<3 \to -3<10-2BC<3 \to 3.5<BC<6.5$. Similarly, we will get the same range for AC. Thus, as both of them are greater than 3.5, the least value WILL BE for the side AB=3. Sufficient.

B.
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Re: If AB=3, what is the shortest side of triangle ABC? (1) AC= [#permalink]  22 Apr 2014, 04:32
mau5 wrote:
blueseas wrote:
If AB=3, what is the shortest side of triangle ABC?

(1) AC=5

(2) The perimeter of ΔABC is 13.

From F.S 1, we get a valid triangle for AC=5 and BC=4, and the smallest side is AB.
Again, for BC=2.1, we get another valid triangle and this time, the smallest side is BC. As we get 2 different answers, Insufficient.

From F.S 2, we know that AC+BC = 13-3 = 10. Now, we know that the difference of 2 sides must be less than the third side of a valid triangle.

Thus, as have |AC-BC|<3. Replacing the value of AC, we get $|10-BC-BC|<3 \to -3<10-2BC<3 \to 3.5<BC<6.5$. Similarly, we will get the same range for AC. Thus, as both of them are greater than 3.5, the least value WILL BE for the side AB=3. Sufficient.

B.

Dear mau5, could you please elaborate on why |AC-BC|<3?

Thanks!
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Re: If AB=3, what is the shortest side of triangle ABC? (1) AC= [#permalink]  22 Apr 2014, 05:02
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jlgdr wrote:
Dear mau5, could you please elaborate on why |AC-BC|<3?

Thanks!
Cheers
J

This is due to the triangle inequality and comes directly from mau5's previous statement:
Quote:
From F.S 2, we know that AC+BC = 13-3 = 10. Now, we know that the difference of 2 sides must be less than the third side of a valid triangle.

I'll explain the sufficiency of (2) a bit differently to see if it helps.

Suppose AB isn't the shortest side, and there is a shorter side BC. So let BC = 3. Then AC must be 7 (since perimeter is 13). A 3-3-7 isn't a valid triangle due to the triangle inequality (try to draw a triangle with sides 3, 3, and 7!).
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Re: If AB=3, what is the shortest side of triangle ABC? [#permalink]  10 May 2014, 10:48
Stat.(1): you can only plug in BC, which, according to the Third Side Rule must be smaller than 8 (the sum of AB and AC) and greater than 2 (their difference). Plug in BC=2.5 and BC becomes the smallest side; plug in BC=7 and AB is the smallest side. Hence, Stat.(1)->IS->BCE.

Stat.(2) tells you BC+AC=10. Plug in BC=5 AC=5 and AB becomes the smallest side. But can you plug in less than 3 (less than AB)? If you plug in BC=3, AC must be 7 and the Third Side Rule is broken (AB+BC<AC). Any BC smaller than 3 would also fail. So AB must be the smallest side. Hence, Stat.(2)->S->B.
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Re: If AB=3, what is the shortest side of triangle ABC? [#permalink]  03 Jun 2014, 22:46
AB = 3 and BC+CA=10...considering the general rules that sum of 2 sides is greater than the 3rd side and difference between 2 sides is less than the third side...what I get are the 2 combinations for BC and CA ... those are 5+5 and 4+6...so we can have triangles 5,5,3 and 3,4,6...

hence, if I am not missing something, answer to this question would be C (I need both the statements to answer it).
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Re: If AB=3, what is the shortest side of triangle ABC? [#permalink]  04 Jun 2014, 02:51
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execnitinsharma wrote:
If AB=3, what is the shortest side of triangle ABC?

(1) AC=5

(2) The perimeter of ΔABC is 13.

AB = 3 and BC+CA=10...considering the general rules that sum of 2 sides is greater than the 3rd side and difference between 2 sides is less than the third side...what I get are the 2 combinations for BC and CA ... those are 5+5 and 4+6...so we can have triangles 5,5,3 and 3,4,6...

hence, if I am not missing something, answer to this question would be C (I need both the statements to answer it).

The question asks what is the shortest side of triangle ABC? In both your examples the shortest side is 3. Isn't it? Also, notice that we are NOT told that the lenghts of the sides are integers, so there could be many more combinations than you wrote.

The point is that from (2) it follows that no side can be less than or equal AB=3, because if any of the sides is, then the third side must be more than or equal to 7, which would be more than the sum of the other two sides (3 and less than or equal 3).

Does this make sense?
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Re: If AB=3, what is the shortest side of triangle ABC? [#permalink]  04 Jun 2014, 10:13
Bunuel wrote:
execnitinsharma wrote:
If AB=3, what is the shortest side of triangle ABC?

(1) AC=5

(2) The perimeter of ΔABC is 13.

AB = 3 and BC+CA=10...considering the general rules that sum of 2 sides is greater than the 3rd side and difference between 2 sides is less than the third side...what I get are the 2 combinations for BC and CA ... those are 5+5 and 4+6...so we can have triangles 5,5,3 and 3,4,6...

hence, if I am not missing something, answer to this question would be C (I need both the statements to answer it).

The question asks what is the shortest side of triangle ABC? In both your examples the shortest side is 3. Isn't it? Also, notice that we are NOT told that the lenghts of the sides are integers, so there could be many more combinations than you wrote.

The point is that from (2) it follows that no side can be less than or equal AB=3, because if any of the sides is, then the third side must be more than or equal to 7, which would be more than the sum of the other two sides (3 and less than or equal 3).

Does this make sense?

Gosh...what was I thinking You are absolutely correct. Thanks!!
Re: If AB=3, what is the shortest side of triangle ABC?   [#permalink] 04 Jun 2014, 10:13
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