hi,
let us see the restriction on third side, if two sides are 5 and 10..
let it be x, so 10-5<x<10+15 OR 5<x<15..

now lets see the statements..

(1) Perimeter of the triangle is a multiple of 5
since perimeter is multiple of 5, perimeter will be 15 or 20 or 25..
within the restrictions, ONLY value can be 10 and perimeter 25..
Suff

(2) the triangle is isoceles
there are two triangles possible...
5,5,10... since th third side does not fall in the range 5<x<10 .. not possible
5,10,10.. possible
Suff

D
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If the lengths of two sides of a certain triangle are 5 and 10, what is the length of the third side of the triangle

(1) Perimeter of the triangle is a multiple of 5
(2) The triangle is isosceles

We can modify the original condition and the question. If we designate the three sides of the triangle as x, we get 10-5<x<10+5, which is same as 5<x<15. There is 1 variable (x) in the original condition. In order to match the number of variables to the number of equations, we need 1 more equation. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that D is the correct answer choice.
For the condition 1), since x=10, the answer is unique and the condition is sufficient.
For the condition 2), from 5:5:10 and 5:10:10, only 5:10:10 is possible. Hence, the answer is unique and the condition is sufficient. Therefore, the correct answer is D.
l For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Never mind, got it!
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Can I translate "5 < x < 15" to "x has to be bigger than the smallest side, and smaller than the sum of the other two sides"??

Does this make sense?

Thanks.

Hi,
X has to be larger than the positive difference between the two other sides and smaller than the sum of the two other two sides.
10-5
In general, x can be smaller than the smallest known side. Eg 10 and 9 allows x to be between 1 and 19.

Not sure if this was what you asked?

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For the first statement how come the third side not be 10√3?

If the triangle has angles 30-60-90 a possible scenario is 1-√3-2 for its sides.
Which will make the triangle to have a perimeter of : 5+10+10√3 = 5(1+2+√3)=5*(3+√3) which IS a multiple of 5.
Why am I wrong here?
Does the perimeter HAVE TO be an integer in order to be a multiple of 5?

On the GMAT, 'multiple of x' refers only to integer multiples. So, 15 is a multiple of 5 (since it's 3*5) but 6 isn't a multiple of 5, even though 6 = 1.2*5.
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The key to solving this problem is the property of triangles: Given any two sides, the third side of the triangle is greater than the difference and lesser than the sum of the two given sides.

)Property of a triangle: The sum of any two sides of a triangle is always greater than the third side of the triangle.
So, the third side of triangle could take any value from 6 to 14(both inclusive).
Statement 1: This is only possible if the third side is also a multiple of 5 because the sum of other two sides is 15 i.e. a multiple of 5.
So, the only possible value for third side is 10. Hence, Sufficient.
Statement 2: Isosceles triangle has two equal sides.
Other 2 sides are 10 and 5. Third side can’t be 5 because then the sum of two sides (5 + 5) = 10(third side).
So, the only possible value of third side is 10. Hence, Sufficient.

Bunuel please tag this as a GMATPrep question, thank you!

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Added the tag. Thank you!
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