In the adjoining figure, `DeltaABC` circumscribes a circle and touched the circle at the points P, Q, R. If `AP =4 cm, BP=6cm, AC =12cm and BC =x` cm, then determine the calue of x.

Explanation

Now `DeltaAOP-=DeltaAOR[becauseOP=OR, angle OPA= angleORA =90^(@)and `hypotenuse OA is common to both.]
`therefore AP= AR` [similar sides of congruent triangles]
Then `CR =AC-AR`
`=12cm -AP [because AR=AP]`
`=12cm -4 cm [because AC=12 cm and AP =4cm]= 8 cm`
Again `Delta COQ -= DeltaCOR ` [by R-H-S condition of congruency]
`therefore CQ = CR` [similar sides of congruent triangles]
`=8 cm [because CR=8 cm]........(1)`
Again `DeltaBOQ -=DeltaBOP` [by the R-H-S condition of congrucy]
`therefore BQ =BP[because` similar sides of congruent triangles]
`=6 cm [BP =6 cm].........(2)`
Then,` BC =BQ+CQ`
`=6 cm +8 cm ` [from (1) and (2)]
`therefore BX =x cm =14cm:`