From an outside point P, two tangents, PA and PB, are drawn to a circle.A third tangent to the circle at a point Q intersects PA and PB at R and S respectively. If PA=10 cm, what is the perimeter of △PRS?

Correct — C. Two tangents drawn from one external point are equal.

Tangents from R: RA = RQ
Tangents from S: SB = SQ
Tangents from P: PB = PA = 10 cm

Split the third side: RS = RQ + QS = RA + SB
Perimeter = PR + RA + SB + SP = PA + PB
= 10 + 10 = 20 cm → option (c)

Why the others are wrong

• (a)
  10 cm — 10 cm is one tangent, PA. The perimeter collects PR + RA = PA and PS + SB = PB, so it is PA + PB.
• (b)
  15 cm — 15 cm would need RS to be shorter than RA + SB. It is not: RS = RQ + QS = RA + SB, because the tangents from R and from S are equal.
• (d)
  30 cm — 30 cm adds RS on top of PA + PB. RS is not extra length: RQ equals RA and QS equals SB, which already lie inside PA and PB.

Equal tangent lengths: from any point outside a circle, the two tangent segments to the points of contact are equal.

Here the rule is used at P, R and S. The third tangent's pieces RQ and QS swap in for RA and SB, so the perimeter of △PRS collapses to PA + PB = 2 × PA, wherever Q sits on the arc facing P.

Key facts

• Tangent segments from an external point to a circle are equal in length.
• The radius to a point of contact is perpendicular to the tangent.
• If a tangent at Q cuts tangents PA and PB at R and S, the perimeter of △PRS is 2 × PA.

Common traps

• Treating RS as a separate unknown and hunting for it, when it splits into RA + SB.
• Answering 10 cm because the stem gives a single length, PA.

No directly related past PYQ was found.