which is not possible with side below?

The correct answer is D. The question asks which set of values cannot be the three sides of the triangle shown, and the test is the triangle inequality: the sum of any two sides must be greater than the third, so it is enough to check whether the two shorter sides together reach the longest side.

• Option A (6, 7, 9): 6+7=13 > 9, 6+9=15 > 7 and 7+9=16 > 6. A valid triangle, so this is possible.
• Option B: The lengths are the same as in A, so it is equally possible.
• Option C (36, 49, 81): 36+49=85 > 81. Valid, and since 85 > 81 the largest angle is acute.
• Option D (216, 343, 729): 216+343=559, which is less than 729. The two shorter sides cannot span the longest side, so no triangle can be formed. This is the impossible set.

Core concept: triangle inequality and feasibility of side lengths. It also underlies constructing triangles from given data, a basic skill in geometry-based reasoning questions.

Trap: Candidates check only one or two pairs, or they miscalculate the squares and cubes. Always test the largest side against the sum of the other two.