SSC CGL 2024 Tier-I · 25 September 2024, Shift 1 · Quantitative Aptitude · Question 23 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 10 September 2026.

If the lengths of two sides of an isosceles triangle are 6 cm and 12 cm, then find the length of the third side.

(a)8 cm
(b)12 cm
(c)14 cm
(d)6 cm

Answer

Why

Correct — B. Isosceles means two sides are equal, so the third side has to repeat one of the lengths already given — 6 cm or 12 cm, nothing else.

Try 6 cm: the sides are 6, 6, 12. Then 6 + 6 = 12, which is not greater than 12, so the triangle inequality fails and the three points lie on one straight line.

Try 12 cm: the sides are 6, 12, 12. Now 6 + 12 = 18 > 12 and 12 + 12 = 24 > 6, so every pair beats the side left out.

The third side is 12 cm → option (b).

Why the others are wrong

(a)
8 cm — 8 cm gives sides 6, 8 and 12 — three different lengths, so the triangle is scalene, not isosceles. It passes the triangle inequality, which is precisely why it tempts.
(c)
14 cm — 14 cm also leaves three unequal sides, 6, 12 and 14. A perfectly valid triangle, but the stem says isosceles and no two of these lengths match.
(d)
6 cm — 6 cm is the near miss. Sides 6, 6 and 12 look isosceles, but 6 + 6 = 12 exactly, so the two short sides fold flat onto the long one and no triangle exists.

Concept

Two conditions have to hold together, and each wrong option fails one of them.

Isosceles pins the unknown side to one of the two given lengths. Any third value makes the triangle scalene, so 8 cm and 14 cm are ruled out before any measuring.

The triangle inequality requires the sum of any two sides to be strictly greater than the third. 6 + 6 = 12 is not strictly greater, so 6, 6, 12 is degenerate: the vertices are collinear and the figure has zero area.

That leaves 6, 12, 12 as the only set satisfying both.

The equal pair does not have to be the shorter pair, and that is where the question bites.

Assume the two printed lengths are themselves the equal sides and you have a triangle with sides 6 and 12 that cannot both be repeated — the stem gives two different lengths, so one is doubled and the other stands alone.

Key facts

In any triangle the sum of two sides is strictly greater than the third side.
Equality, as in 6 + 6 = 12, gives a degenerate triangle of zero area rather than a triangle.
An isosceles triangle has at least two equal sides, so the unknown side must repeat a given length.
From lengths 6 cm and 12 cm the only valid isosceles set is 6, 12, 12.

Study next

The range form of the triangle inequality, where the third side lies strictly between |a − b| and a + bClassifying a triangle as acute, right or obtuse by comparing a² + b² with c²Perimeter and area of an isosceles triangle from its two distinct side lengths

Common traps

Treating any length that looks reasonable, such as 8 cm or 14 cm, as admissible
Accepting 6, 6, 12 because it is isosceles, without checking whether 6 + 6 > 12
Using the inequality in its non-strict form, sum ≥ third side, which lets the degenerate case through

Two lengths and the word isosceles, with the degenerate pairing sitting there as an option — the whole item turns on the strictness of the triangle inequality.

Isosceles triangles are also worked at Quant Q.10 of this paper, where AC = BC fixes two equal base angles, and at 9 Sep 2024, 12:30, Quant Q.13, a right isosceles triangle of area 50 square units.

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