In the given figure, PT and PS are two tangents to a circle with centre O, such that PT=5 cm and ∠TPS=60∘. Find the length of chord TS.

In the given figure, PT and PS are two tangents to a circle with centre O, such that

PT = 5 cm

and

∠ TPS = 60^{\circ}

As we know that the tangents drawn from an external point to the circle are equal in length.
Thus,

PT = PS

.
Also, angles opposite to equal sides are equal.
Thus,

∠ P T S = ∠ P S T

Now, using angle sum property in a triangle in

ΔPTS

∠ P T S + ∠ P S T + ∠ T P S = 180 °

∠ P T S + ∠ P S T + 60 ° = 180 °

∠ P T S + ∠ P T S = 180 ° − 60 °

2 ∠ P T S = 120 °

∠ P T S = 60 °

Therefore,

∠ P T S = 60 ° = ∠ P S T

.
Thus, PTS is an equilateral triangle.
So, all sides of triangle PTS are equal and thus

TS = PT = 5 cm

.
Therefore, option 1 is correct.

In the given figure, PT and PS are two tangents to a circle with centre $O$ , such that $PT = 5 cm$ and $∠ TPS = 60^{\circ}$ . Find the length of chord TS.