Is the triangle ABC with sides of length a, b, c an isosceles triangle? (I) a^2 + b^2 = c^2 (II) (a + b - c)^2 = (b + c - a)^2

1. (I) a^2+b^2=c^2 only tells us the triangle is right-angled; sides could be 3,4,5 (not isosceles). 2. (II) Take square roots (sides are positive lengths): |a+b-c| = |b+c-a|. Expanding both gives a+b-c = b+c-a or a+b-c = -(b+c-a). 3. First case: 2a = 2c, so a = c — isosceles. Second case: a+b-c = a-b-c, giving b = 0, impossible for a triangle. 4. So (II) alone confirms a = c (isosceles). (I) alone does not. Answer: (II) alone is sufficient. _Source: TSICET 2024 PYQ Q.2 (Shift 1, Data Sufficiency)_