SSC CGL 2024 Tier-I · 24 September 2024, Shift 3 · Quantitative Aptitude · Question 2 of 99

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.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

Let ABC be a triangle. If the mid-points of AB, BC and AC are D, E and F, respectively, then ∆DEF is NOT congruent to _____.

(a)∆BDE
(b)∆ADF
(c)∆ECF
(d)∆ABC

Answer

Why

Correct — D.

D, E, F are the midpoints of AB, BC, AC, so the midpoint theorem gives
DE = ½AC, EF = ½AB, DF = ½BC.

Every side of ∆DEF is half the matching side of ∆ABC, so the two are similar in the ratio 1 : 2 — similar, not congruent.

The three corner triangles carry that same set of lengths: ∆ADF, ∆BDE and ∆ECF each have sides ½AB, ½BC, ½AC, so each is congruent to ∆DEF by SSS.

The triangle ∆DEF is NOT congruent to is ∆ABC → option (d).

Why the others are wrong

(a)
∆BDE — BD = ½AB, BE = ½BC, DE = ½AC — the same three lengths as ∆DEF, whose sides are EF = ½AB, DF = ½BC and DE = ½AC. Congruent by SSS.
(b)
∆ADF — AD = ½AB, AF = ½AC, DF = ½BC matches ∆DEF side for side. The medial triangle cuts ∆ABC into four identical pieces, and this is one of them.
(c)
∆ECF — EC = ½BC, CF = ½AC, EF = ½AB is the ∆DEF triple again, so ∆ECF ≅ ∆DEF by SSS.

Concept

Joining the midpoints of the three sides cuts a triangle into four congruent triangles, each similar to the original.

The midpoint theorem does the work: the segment joining the midpoints of two sides is parallel to the third side and half its length. So DE, EF and DF are half of AC, AB and BC, and each corner triangle inherits the same three half-lengths.

∆DEF and ∆ABC therefore stand in the ratio 1 : 2, which is similarity, not congruence. Areas follow the square of that ratio, so each of the four pieces is one quarter of ∆ABC.

The word NOT is capitalised in the paper because three of the four options are congruent to ∆DEF; the odd one out is the parent triangle.

Key facts

Midpoint theorem: the segment joining the midpoints of two sides is parallel to the third side and half its length.
The medial triangle divides a triangle into four congruent triangles, each of area one quarter of the whole.
∆DEF and ∆ABC are similar in the ratio 1 : 2, so congruent is the wrong word for that pair.

Study next

Midpoint theorem and its converseSimilar triangles: the ratio of areas is the square of the ratio of sidesSSS, SAS, ASA and RHS congruence criteria

Common traps

Treating similar as congruent because the shapes match.
Missing the capitalised NOT and picking a triangle that is congruent.
Assuming the middle triangle differs from the three corner ones when all four are congruent.

The midpoint theorem is also cashed out as a length or an area rather than as congruence.
25 Sep 2024, 16:00, Quant Q.25 asks for ST when S and T are midpoints of PQ and PR and QR = 12 cm; 11 Sep 2024, 12:30, Quant Q.5 asks for the area of quadrilateral MNRQ inside a 46 cm² triangle.

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