The sides of four triangles are given below: (i) 20 cm, 22 cm, 24 cm (ii) 15…

For any triangle with sides a, b, and the longest side c, the converse of the Pythagorean theorem states that the triangle is right-angled — with the right angle opposite side c — if and only if a2 + b2 = c2. Checking whether the sum of the squares of the two shorter sides equals the square of the longest side is a quick way to test each option without measuring angles directly.

1. 20 cm, 22 cm, 24 cm: 202 + 222 = 400 + 484 = 884, but 242 = 576. Since 884 is not equal to 576, this set does not satisfy the relation (the triangle is acute, not right-angled).

2. 15 cm, 32 cm, 37 cm: 152 + 322 = 225 + 1024 = 1249, but 372 = 1369. Since 1249 is not equal to 1369, this set does not satisfy the relation (the triangle is obtuse, not right-angled).

3. 11 cm, 60 cm, 61 cm: 112 + 602 = 121 + 3600 = 3721, and 612 = 3721. Since 3721 equals 3721, this set satisfies the relation exactly.

4. 19 cm, 40 cm, 41 cm: 192 + 402 = 361 + 1600 = 1961, but 412 = 1681. Since 1961 is not equal to 1681, this set does not satisfy the relation (the triangle is acute, not right-angled).

As an independent check, the 11-60-61 triple can also be generated from the standard odd-integer identity for Pythagorean triples: for an odd number m greater than or equal to 3, the triple (m, (m2 - 1)/2, (m2 + 1)/2) always satisfies the Pythagorean relation. With m = 11: (m2 - 1)/2 = (121 - 1)/2 = 60 and (m2 + 1)/2 = (121 + 1)/2 = 61, which independently reproduces 11, 60, 61 — confirming the direct calculation above.

Therefore, only the triangle with sides 11 cm, 60 cm, and 61 cm is a right triangle.