Is the perimeter of triangle T greater than the perimeter of

Hi dallasgmat,

Note:
- In triangles, sum of any two sides is GREATER than the third side.
- In triangles, difference of any two sides is LESS than the third side.

Keep in mind the above to avoid concept mistakes.

cheers,
Jack

Is the perimeter of triangle T greater than the perimeter of square S ?
(1) T is an isosceles right triangle.
(2) The length of the longest side of T is equal to the length of a diagonal of S.

Solution:
Bunnuel is right the ans is C.
(1) No information about S is given, so insufficient
(2) Let the one side of s = x, so Diagonal of s = x√2
Perimeter of S = 4x
Thus, the longest side of T = x√2
No other information is given about other sides of T, So insufficient.

Considering C
the sides of T = x, x, x√2 [Sides of triangle with 90 degree, 45 degree and 45 degree is s, s, s√2]
Perimeter of T = x + x + x√2 which is Less than 4x.
Ans. C

Let a, b, and c denote the three sides of triangle T, with c as the largest side, and let s denote a side of square S.
The question then becomes: is \((a+b+c)>4s\)?

(1) Does not give any information about the square S. Thus, Not Sufficient.

(2) \(c = s\sqrt{2}\).
From the triangle property, we know that sum of lengths of any two sides of a triangle is always greater than the length of third side. Thus, for triangle T, we have:
\(a+b>c\)
\(a+b+c>2c\)
\(a+b+c> 2\sqrt{2}s\)
This still is insufficient data, as we cannot prove weather
\(4s>(a+b+c)>2\sqrt{2}s\) OR
\((a+b+c)>4s\) (true in case of equilateral triangles)

(1)+(2) a = b, and \(c = a\sqrt{2} = s\sqrt{2}\)
Thus, we get a = s.
\(a+b+c = a(2+\sqrt{2}) < 4s\). Thus, we can have our answer to the question: is (a+b+c)>4s? No!

So correct answer is (C): (1) and (2) are sufficient together, but not alone..

Bunnel thanks for the explanation

Again! Well done explained! Couldn´t figure it out myself!

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is the perimeter of triangle T greater than the perimeter of square S?

(1) T is an isoceles right triangle.
(2) The length of the longest side of T is equal to the length of a diagonal of S.

There are 3 variables (3 sides) in a triangle and 1 (one side) in a square. We need 4 equations when only 2 equations are given by the conditions, so there is high chance (E) will be our answer.
Looking at the conditions together,
perimeter of T < perimeter of S, so the answer is 'no' and this is sufficient. The answer becomes (C).

For cases where we need 3 more equation, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

Hi Bunuel, Thanks for explanation.

I have one question, had the question asked Is the Area of triangle T greater than the Area of square S ?

Statement 2 alone would have sufficient, because for a given perimeter, equilateral triangle has the max area,
=> Area of equilateral triangle with side \sqrt{2} * s = {\sqrt{3} / 4} * (\(4 * s^2\)} = \sqrt{3} / 2 * \(s^2\) < \(s^2\)(area of square).

am i right?

Bunuel Is it not the other way around? Is not the perimeter smallest for an equilateral triangle?

Bunuel GMATBusters

Could you advise how to compare a perimeter of 45-45-90 vs an equilateral triangle?

My alternative approach:
I need measures of all 3 sides of triangle and length of a side of square.
St1 : no info about square side insuff,
St 2: I can get side of square from diagonal and the longest square side of triangle which could be hypotenuse but if triangle is 30-60-90 and other two shorter lengths will be different from 45-45-90 Insff.

Combine: now I know triangle is 45-45-90 , since one side (hyp) is known from square diagonal, I can find remaining two sides corr to 45 angle, Suff

you are right:

St 2: I can get side of square from diagonal and the longest square side of triangle which could be hypotenuse but if triangle is 30-60-90 and other two shorter lengths will be different from 45-45-90 Insff.
in fact, if we know only one side, infinite types of triangle are possible and hence Perimeter.

Your Query: Could you advise how to compare a perimeter of 45-45-90 vs an equilateral triangle?
to compare we need the relation between side of equilateral traingle and at least one side with the corresponding angle for 45-45-90

Hope, it clarifies your query, feel free to tag again for any issue.

Hi everyone

This is a good Data Sufficiency question.

Question stem: Is the perimeter of triangle T greater than the perimeter of square S?

(1) T is an isoceles right triangle.

Does this tell us anything about the parameters of the square? No.
So, this statement is not enough to answer the question.

Thus, answer can not be (A) or (D).

(2) The length of the longest side of T is equal to the length of a diagonal of S.

We know the length of one side of triangle is equal to diagonal of the square.
However, we don't know anything about the other 2 sides of the triangle.

Thus, this statement is not enough to answer the question.

So, answer can not be (B).

Let's consider statement 1 and 2 together:
If the length of longest side of triangle is x, perimeter of triangle will be \(x+ (\sqrt{2})x\)
Likewise, perimeter of square S with diagonal x shall be \(2(\sqrt{2})x\)

Clearly, perimeter of S is greater.

Thus, answer is (C).

Was I clear and elaborate? Would anyone like me to explain any specific point?

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