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# What is the perimeter of isosceles triangle ABC?

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What is the perimeter of isosceles triangle ABC? [#permalink]

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01 Sep 2006, 00:41
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What is the perimeter of isosceles triangle ABC?

(1) The length of side AB is 9

(2) The length of side BC is 4
[Reveal] Spoiler: OA

Last edited by Bunuel on 19 Jul 2012, 02:36, edited 1 time in total.
Edited the question and added the OA.

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01 Sep 2006, 01:34
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C

AB = 9 units
BC = 4 units
AC will have to be equal to 9 units. It cannot be 4 units because the length of the sum of two sides of a triangle must be greater than the third side and the problem states that the traingle is an isosceles triangle.

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Re: What is the perimeter of isosceles triangle ABC? (1) The [#permalink]

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18 Jul 2012, 23:15
(1) AB = 9. We only know the length of one side. Not sufficient.
(2) BC = 4. We only know the length of one side. Not sufficient.

(1)+(2) If AB=AC=9 and BC=4 then the perimeter would be 9+9+4=22 but if AB=9 and AC=BC=4 then the perimeter would be 9+4+4=17. Not sufficient.

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Re: What is the perimeter of isosceles triangle ABC? (1) The [#permalink]

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19 Jul 2012, 02:50
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monirjewel wrote:
(1) AB = 9. We only know the length of one side. Not sufficient.
(2) BC = 4. We only know the length of one side. Not sufficient.

(1)+(2) If AB=AC=9 and BC=4 then the perimeter would be 9+9+4=22 but if AB=9 and AC=BC=4 then the perimeter would be 9+4+4=17. Not sufficient.

Answer to this question should be C, not E.

What is the perimeter of isosceles triangle ABC?

(1) The length of side AB is 9. Clearly insufficient.

(2) The length of side BC is 4. Clearly insufficient.

(1)+(2) We know the lengths of the two sides of isosceles triangle ABC: AB=9 and BC=4, hence the length of AC is either 4 or 9. Relationship of the Sides of a Triangle: The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Now, according to this, AC cannot equal to 4, because in this case the length of AB would be greater than the sum of the other two sides, AC and BC, (AB=9>AC+BC=4+4=8), hence AC=9 and P=9+9+4=22. Sufficient.

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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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06 Sep 2013, 04:25
Now it is clear to me. Thanks
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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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13 Aug 2014, 06:39
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Why is a side less than the sum of two sides? Check this figure out !
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TriangleInequality.svg.png [ 46.39 KiB | Viewed 15092 times ]

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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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11 Jul 2015, 02:11
Bunuel wrote:
(1)+(2) We know the lengths of the two sides of isosceles triangle ABC: AB=9 and BC=4, hence the length of AC is either 4 or 9. Relationship of the Sides of a Triangle: The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Now, according to this, AC cannot equal to 4, because in this case the length of AB would be greater than the sum of the other two sides, AC and BC, (AB=9>AC+BC=4+4=8), hence AC=9 and P=9+9+4=22. Sufficient.

Could I also say that given that AB = 9 and BC = 4 the third side can not be equal to 4 as 9 (AB) - 4 (BC) = 5 and then the third side would be smaller than the sum of the two others?

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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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11 Jul 2015, 02:31
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noTh1ng wrote:
Bunuel wrote:
(1)+(2) We know the lengths of the two sides of isosceles triangle ABC: AB=9 and BC=4, hence the length of AC is either 4 or 9. Relationship of the Sides of a Triangle: The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Now, according to this, AC cannot equal to 4, because in this case the length of AB would be greater than the sum of the other two sides, AC and BC, (AB=9>AC+BC=4+4=8), hence AC=9 and P=9+9+4=22. Sufficient.

Could I also say that given that AB = 9 and BC = 4 the third side can not be equal to 4 as 9 (AB) - 4 (BC) = 5 and then the third side would be smaller than the sum of the two others?

Yes, that's correct:
(9 - 4) < (third side) < (9 + 4)
5 < (third side) < 13.
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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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19 Dec 2015, 22:06
Sum of two sides should be greater than the third side
Difference of two sides should be less than the third sides.

Statement 1 gives a side length but puts into question whether this is one of the two same sized sides of an isosceles triangles.
Statement 2 follows the same path.

Using the two statements and applying the two properties about sum and difference of two sides, we can deduce that 4 cannot be one of the two equal sides. THe two equal sides measure 9. Hence C.
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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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28 Dec 2015, 03:35
Brunel, is there a formula to find Area of an isosceles triangle if we are given only the Perimeter of that Triangle ?

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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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03 Mar 2017, 13:30
Posting official solution of this problem.
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solution.PNG [ 109.75 KiB | Viewed 4942 times ]

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Re: What is the perimeter of isosceles triangle ABC? [#permalink]

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04 Mar 2017, 06:30
ABC is an iscoceles triangle
St 1: AB =9. 2nd side will be 9 or other value which is equal to third side. INSUFFICIENT
St 2: BC =9. 2nd side will be 4 or other value which is equal to third side. INSUFFICIENT

St 1 & St 2: AB = 9 and BC = 4, AC could be 9 or 4. but it cannot be 9 as it violates the theorem: sum of two sides of the triangle is greater the the third side.
Hence AC has to be 9. ANSWER.

Option C

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What is the perimeter of isosceles triangle ABC? [#permalink]

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21 Mar 2017, 00:47
GMATT73 wrote:
What is the perimeter of isosceles triangle ABC?

(1) The length of side AB is 9

(2) The length of side BC is 4

OFFICIAL SOLUTION

The perimeter of a triangle is equal to the sum of the three sides.

(1) INSUFFICIENT: Knowing the length of one side of the triangle is not enough to find the sum of all three sides.

(2) INSUFFICIENT: Knowing the length of one side of the triangle is not enough to find the sum of all three sides.

Together, the two statements are SUFFICIENT. Triangle ABC is an isosceles triangle which means that there are theoretically 2 possible scenarios for the lengths of the three sides of the triangle:

(1) AB = 9, BC = 4 and the third side, AC = 9 OR
(1) AB = 9, BC = 4 and the third side AC = 4.

These two scenarios lead to two different perimeters for triangle ABC, HOWEVER, upon careful observation we see that the second scenario is an IMPOSSIBILITY. A triangle with three sides of 4, 4, and 9 is not a triangle. Recall that any two sides of a triangle must sum up to be greater than the third side. 4 + 4 < 9 so these are not valid lengths for the side of a triangle.

Therefore the actual sides of the triangle must be AB = 9, BC = 4, and AC = 9. The perimeter is 22.

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What is the perimeter of isosceles triangle ABC?   [#permalink] 21 Mar 2017, 00:47
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