Question

From a point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP = 2r, show that Triangle TPS is equilateral triangle. ​

Answer 1

Answer:

Triangle TPS is equilateral

Step-by-step explanation:

In  ΔOTP,

OT ⊥ PT (∵ PT is a tangent to the circle)

Using trigonometry in this triangle,

cos( ∠TOP ) = $\frac{OT}{OP}$

OT = r (Radius) and OP = 2r (Given)

∴ cos ( ∠TOP ) = 1/2

⇒ ∠TOP = 60°

∴ ∠TPO = 90° - ∠TOP = 30°

Similarly,

∠SPO = 30°

∴ ∠TPS = 60°

We know that TP = TS

∴ ∠PTS = ∠ PST = α

In ΔTPS,

2α + 60° = 180°

⇒α = 60°

Hence all the angles in ΔTPS are equal

∴ It's an equilateral triangle

This is a trigonometric approach to this question. You can also solve it using similar triangles. Make sure to appreciate the answer if it helped

Answer 2

Answer:

Triangle TPS is equilateral

Step-by-step explanation:

In ΔOTP,

OT ⊥ PT (∵ PT is a tangent to the circle)

Using trigonometry in this triangle,

cos( ∠TOP ) = \frac{OT}{OP}

OP

OT

OT = r (Radius) and OP = 2r (Given)

∴ cos ( ∠TOP ) = 1/2

⇒ ∠TOP = 60°

∴ ∠TPO = 90° - ∠TOP = 30°

Similarly,

∠SPO = 30°

∴ ∠TPS = 60°

We know that TP = TS

∴ ∠PTS = ∠ PST = α

In ΔTPS,

2α + 60° = 180°

⇒α = 60°

Hence all the angles in ΔTPS are equal

∴ It's an equilateral triangle

This is a trigonometric approach to this question. You can also solve it using similar triangles. Make sure to appreciate the answer if it helped