Question

In the given figure, PA is the tangent to the circle with centre O such that OA = 10 cm, AB = 8 cm and AB ⊥ OP. Find the length of PB.

Answer 1

Given: OA = 10 cm

AB = 8 cm

In Δ OAB,

In right-angled triangle Δ OAB,

∠ OAB = 90°

Since AB ⊥ OP (AB is perpendicular to OP)

Using Pythagoras theorem:

OA² = OB² + AB²   

10² = OB² + 8²

100 = OB² + 64

OB² = 100 − 64

OB² = 36

OB = `sqrt36`

OB = 6 cm

In △OAB and △OAP,

∠ AOB = ∠ POA   ...(Common angle)

Since AB ⊥ OP (AB is perpendicular to OP)

△OAB ∼ △OAP   ...(by AA similarity criterion)

From the similarity, the ratio of corresponding sides is equal:

`(OA)/(OB) = (OP)/(OA)`

`10/6 = (OP)/10`

`OP = (10 xx 10)/6`

`OP = 100/6`

`OP = 50/3`

Since AB ⊥ OP, and B lies on OP,

The length of PB = OP − OB

= `50/3 - 6`

= `(50 - 18)/3`

= `32/3` cm