Question

PA and PB are two tangents drawn from an external point P to a circle with Centre O and radius 4 cm if PA perpendicular PB find the length of Each tangent​

Answer 1

Answer:

CA is perpendicular to AP and CB is perpendicular to BP  

Again AC = BC = 4 (radius of the circle)

Also AP = PB = (Tangents from point P)

So, BPAC is a square.

=> AP = PB = BC = CA = 4 cm

So, length of tangents are 4 cm each

I hope this helps!

Answer 2

Diagram:-

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Given:

• OA=OB=4cm (Radius)

• ∠APB=90°

• AP & PB=Tangents

To find:

• Length of Tangents

Solution:

⇒ ∠PAO=∠PBO=90°(angle made by Tangent with radius)

Now,

⇒ ∠APB+∠PBO+∠PAO+∠AOB=360°(Sum of all ∠ of quadrilateral)

⇒ 90°+90°+90°+∠AOB=360°

⇒ 270°+∠AOB=360°

⇒ ∠AOB=360°-270°

⇒ ∠AOB=90°

Now join AB in diagram

~Applying Pythagoras theorem in ∆AOB

⇒ AO²+OB²=AB²

⇒ (4cm)²+(4cm)²=AB²

⇒ 16cm²+16cm²=AB²

⇒ 32cm²=AB

⇒ √32cm²=AB

⇒ 4√2cm=AB

Now assume length of each tangent be x

Since, PAB is a right angled ∆, we can apply Pythagoras theorem in it.

~Applying Pythagoras theorem in ∆PAB

⇒ PA²+PB²=(4√2cm)²

⇒ x²+x²=32cm²

⇒ 2x²=32cm²

⇒ x²=32cm²/2

⇒ x²=16cm²

⇒ x=√16cm²

⇒ x=4cm

So the required length of tangent is 4cm.