subhajeet wrote:

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.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Can anyone please provide a solution for this.

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..

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