(i) From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at the point E and PA = 14 cm, find the perimeter of PCD.

(ii) If PA = 11cm, PD = 7 cm, then DE = ?

i) From P we have tangents PA and PB

Hence PA = PB …tangents from same point are equal …(a)

Point C is on PA

From C we have tangents CA and CE

⇒ CA = CE …tangents from same point are equal …(i)

Point D is on PB

From D we have two tangents DE and DB

⇒ DE = DB … tangents from same point are equal …(ii)

Consider ΔPCD

⇒ perimeter of ΔPCD = PC + CD + PD

From figure CD = CE + ED

⇒ perimeter of ΔPCD = PC + CE + ED + PD

Using (i) and (ii)

⇒ perimeter of ΔPCD = PC + CA + DB + PD

From figure we have

PC + CA = PA and DB + PD = PB

⇒ perimeter of ΔPCD = PA + PB

Using (a)

⇒ perimeter of ΔPCD = PA + PA

⇒ perimeter of ΔPCD = 2(PA)

PA is 14 cm given

⇒ perimeter of ΔPCD = 2 × 14

⇒ perimeter of ΔPCD = 28 cm

ii) PA = 11 cm …given

using (a)

PB = 11 cm

From figure

⇒ PB = PD + DB

Using (ii)

⇒ PB = PD + DE

⇒ 11 = 7 + DE …PD is 7 cm given

⇒ DE = 5 cm

Hence DE = 5 cm