Question

in the given figure from a point P, two tangents PT and PS are drawn to a circle with centre O such that angle SPT =120 degree prove that OP =2PS

Answer 1

Consider ΔOPS and ΔOPT

OS = OT ( radii)

∠OSP = ∠OTP = 90 (tangents are perpendicular to the radii)

SP = ST ( tangents to a circle from the external point are congruence)

ΔOPS ≅ ΔOPT ( By SAS criterion)

The corresponding parts of the corresponding triangles are congruent.

∠OPS = ∠OPT

since ∠SPT = 120° and ∠OPS = ∠OPT

we have ∠OPS = ∠OPT = 60°

∠POS = ∠POT = 30°

Consider In a ΔPOS

sin 30° = PS / OP

1 / 2 = PS / OP

OP = 2PS.

Answer 2

Given,

In the given figure from a point P, two tangents PT and PS are drawn to a circle with centre O such that ∠SPT = 120°

To Find,

Prove that OP = 2PS

Solution,

Consider ΔOPS and ΔOPT

OS = OT ( radii)

∠OSP = ∠OTP = 90 (tangents are perpendicular to the radii)

SP = ST ( tangents to a circle from the external point are congruence)

ΔOPS ≅ ΔOPT ( By SAS criterion)

The corresponding parts of the corresponding triangles are congruent.

∠OPS = ∠OPT

since ∠SPT = 120° and ∠OPS = ∠OPT

we have ∠OPS = ∠OPT = 60°

∠POS = ∠POT = 30°

Consider In a ΔPOS

sin 30° = PS / OP

1 / 2 = PS / OP

OP = 2PS.

Hence, OP = 2PS.