What is the perimeter of isosceles triangle ABC?

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.

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.

Answer: C.

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Hope it helps.
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(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: E.

Now it is clear to me. Thanks

Why is a side less than the sum of two sides? Check this figure out !

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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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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Brunel, is there a formula to find Area of an isosceles triangle if we are given only the Perimeter of that Triangle ?

Thanks in Advance

Posting official solution of this problem.

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

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.

The correct answer is C.
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Hii
There is NO formula/way to find area of isosceles triangle because Area is not constant.

The Area is not unique, if only perimeter of an isosceles triangle is given.

For example:
if sides of a triangle are 9,9,4 (Perimeter = 22) ,area =17.54
if sides of a triangle are 8,8,6 (Perimeter = 22), area = 22.64

How to find area of an isosceles triangle with sides b,b,a



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(1) The length of side AB is 9 - Clearly NS.
(2) The length of side BC is 4 - Clearly NS.

1)+2):

The length of the third side say x will be between (9-5) < x > (9+5) -> 4<x>14.
So, clearly the third side cannot be 4, it has to be >4 and <9. Knowing that the triangle is isosceles we are left with considering the third side as 9.

Perimeter = 9+9+4 = 22

Answer: (C).

Note ans: C
can anyone tell me what is the difference between this problem above and the next one:

What is the perimeter of isosceles triangle MNP?

(1) MN = 16
(2) NP = 20
Note : ans :E

+
if i'm right :
in the first case we can find on probability : 4+9=13, but not 4+4<9
in the second case : we get two probabilities :
16+20>16, 16+16>20
Is it right?

Hello

I think you have already answered your own question in your next comment. In case of MNP triangle problem, when we combine the two statements, the two possible triangles are:
16, 16, 20 and
20, 20, 16

Now both the above triangles satisfy triangle inequality (sum of smaller two sides greater than third side), so both triangles are possible. Hence we could have two different perimeters: 16+16+20 AND 20+20+16
Thats why answer should be E

To find the perimeter of an isosceles triangle we need atleast the length of two unequal sides such that twice the length of smaller side is greater than the larger side.
Statement 1. Length of 1 side of an isosceles triangle is given but we are still unaware of the length of the side that is not equal to AB. Hence, Insufficient.
Statement 2. Statement 2 also mentions the length of just 1 side. Hence, Insufficient.
Statement 1 & 2 together. AB = 9, BC = 4
Third side i.e. AC can’t be 4 because the sum of BC and CA should be greater than AB.
So, AC = 9.
Hence, the perimeter = 9 + 9 + 4 = 22. Hence, Sufficient.

9 + 4 + 4 is not possible given the triangle condition.