Question

PA and PB are tangents of a circle. O is the center. prove angle P + angle O is 180°​

Answer 1

Answer:

Given: O is the centre of the circle. PA and PB are tangents drawn to a circle and ∠APB = 120°.

To prove: OP = 2AP

Proof:

In ΔOAP and ΔOBP,

OP = OP    (Common)

∠OAP = ∠OBP  (90°) (Radius is perpendicular to the tangent at the point of contact)

OA = OB  (Radius of the circle)

∴ ΔOAP is congruent to ΔOBP (RHS criterion)

∠OPA = ∠OPB = 120°/2 = 60° (CPCT)

In ΔOAP,

cos∠OPA = cos 60° = AP/OP

Therefore, 1/2 =AP/OP

Thus, OP = 2AP

Hence, proved.

Step-by-step explanation:

Answer 2

<B=90

<A=90

as radius is perpendicular to tangent.

< B+ < A + <O + < P = 360 ( the sum of angels in a quadrilateral is 360)

90+90+ <O + <P = 360

<O + < P = 360- 180

<O + <P = 180

Hence Proved...