SSC CGL 2025 Tier-I · 17 September 2025, Shift 2 · Quantitative Aptitude · Question 22 of 99

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 13 September 2026.

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Two chords AB and CD intersect at a point P inside a circle. If AP = 6 cm, PB = 8 cm, and CP = 4 cm, what is the length of PD?

(a)10 cm
(b)12 cm
(c)16 cm
(d)18 cm

Answer

Why

Correct — B. When two chords cross inside a circle, the products of their pieces are equal.

Intersecting chords: AP × PB = CP × PD
Left side: 6 × 8 = 48
Substitute: 4 × PD = 48
Divide by 4: PD = 12 cm → option (b)

Why the others are wrong

(a)
10 cm — PD = 10 cm gives CP × PD = 40, short of AP × PB = 48. The two products must match exactly.
(c)
16 cm — PD = 16 cm gives CP × PD = 64, which overshoots AP × PB = 6 × 8 = 48 by 16.
(d)
18 cm — PD = 18 cm gives CP × PD = 72, which would pair with pieces of 6 and 12 on chord AB, not 6 and 8.

Concept

Intersecting chords theorem: if chords AB and CD meet at P inside a circle, AP × PB = CP × PD.

The reason is similarity. △APC and △DPB have equal vertical angles at P, and ∠CAB = ∠CDB because both stand on arc CB. So AP : DP = CP : BP, which cross-multiplies to the rule.

Check: AB = 6 + 8 = 14 cm and CD = 4 + 12 = 16 cm, and each chord's pieces multiply to 48.

The rule extends outside the circle: for two secants from an outside point T, TA × TB = TC × TD, and a tangent from T replaces its product with a square. Quant Q.18 (17 Sep 2025, 12:30) uses that tangent form.

Key facts

Intersecting chords: AP × PB = CP × PD.
Two secants from an outside point T: TA × TB = TC × TD, each measured from T.
A tangent TX from the same point satisfies TX² = TA × TB.

Study next

Chord pieces given as ratios, such as AE : EB = 2 : 3Tangent–secant theorem from a point outside the circleThe perpendicular from the centre bisecting a chord

Common traps

Setting up a ratio instead of a product, AP : PB = CP : PD, which gives 16⁄3 cm.
Multiplying across the chords, AP × CP = PB × PD, which gives 3 cm.

The same product decides 11 Sep 2024, 09:00, Quant Q.11: MO × ON = 9 × 5 = 45 = OP × 6, so OP = 7.5 cm.

16 Sep 2025, 12:30, Quant Q.23 gives the pieces as ratios: AE : EB = 2 : 3 and CE : ED = 5 : 2 make 6x²⁄25 = 10y²⁄49, so x²⁄y² = 125⁄147.

Related PYQs

No directly related past PYQ was found.

Topic: Circle and its chords, tangents, angles subtended by chords of a circle, common tangents to two or more circles — all 140 SSC CGL questions on it, every solved shift.