SSC CGL 2024 Tier-I · 24 September 2024, Shift 3 · Quantitative Aptitude · Question 20 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.

In ΔABC, AB = 6 cm, BC = 9 cm and AC = 12 cm. If AD is the angle bisector of ∠BAC, where D is a point on BC, the

(a)7
(b)3
(c)6
(d)9

Answer

Why

Correct — C. An internal bisector cuts the opposite side in the ratio of the two sides that meet at that angle.

Angle-bisector theorem: BD ⁄ DC = AB ⁄ AC = 6 ⁄ 12 = 1 ⁄ 2
So BC divides into 1 + 2 = 3 equal parts, each 9 ⁄ 3 = 3 cm

BD = 1 × 3 = 3 cm
DC = 2 × 3 = 6 cm

The key marks 6 cm, the piece lying against the longer side AC → option (c).

Why the others are wrong

(a)
7 — 7 is near the length of the bisector AD itself, which is √(AB × AC − BD × DC) = √(72 − 18) = √54 ≈ 7.35 cm. That is a different segment, and it is not a whole number here.
(b)
3 — 3 cm is BD, the piece beside the shorter side AB. The 1 : 2 split puts the small part against the small side, and the key marks the other part.
(d)
9 — 9 cm is BC itself, the undivided side. D lies strictly between B and C, so neither piece can be as long as the whole.

Concept

The angle-bisector theorem says the foot of an internal bisector splits the opposite side in the ratio of the sides enclosing the bisected angle: BD ⁄ DC = AB ⁄ AC.

Read the ratio in the right direction. BD touches B, and the side from A through B is AB — so AB goes on top with BD. Get the direction wrong and you hand back 6 and 3 the other way round.

Once the ratio is 1 : 2, the side of 9 cm is cut into three units of 3 cm. The larger piece always lies against the longer of the two enclosing sides, which is a fast way to check your own answer.

The stem for this row is an image and it is cut short at "where D is a point on BC, the" — the final clause naming the segment is missing from the scan. The configuration is still complete: AB = 6, AC = 12, BC = 9 force BD = 3 and DC = 6, and the key of 6 identifies DC. Do not rely on the picture for the wording.

Key facts

The internal bisector of angle A meets BC at D with BD ⁄ DC = AB ⁄ AC.
With AB = 6 cm, AC = 12 cm the ratio is 1 : 2, so BC = 9 cm splits into 3 cm and 6 cm.
The bisector's own length is AD = √(AB × AC − BD × DC) = √54 cm, roughly 7.35 cm.
Sides 6, 9 and 12 satisfy the triangle inequality, so the triangle described really exists.

Study next

The external bisector and the ratio it cuts on BC producedThe incentre as the common point of the three internal bisectorsStewart's theorem for the length of any cevian

Common traps

Answering BD = 3 when the segment wanted is the one beside AC.
Writing the ratio as BD ⁄ DC = AC ⁄ AB, which swaps the two pieces.
Assuming the bisector also bisects BC, which is true only when AB = AC.

The same theorem is set with different lettering at 12 Sep 2024, 12:30, Quant Q.4 — in ΔDEF the bisector of ∠D meets EF at N with DE = 36 cm, DF = 40 cm and EF = 38 cm, and what is asked for is NF, again the piece beside the longer side.

Related PYQs

No directly related past PYQ was found.