In the given figure, PA and PB are tangents to a circle from an external point P such that PA=4 cm and `angleBAC=135^(@).` Find the length of dhord AB.

Explanation

We have,
`angleBAC=135^(@)`
`:." "anglePAB=180^(@)-135^(@)=45^(@)" "`(L.P.A)...(1)
But since `PA=PB=4`(length of tangents from an external point are equal) …(2)
`implies" "anglePBA=anglePAB" "`(angles opposite to equal sides are equal)
`implies" "anglePBA=45^(@)" "`[from (1)]
Now, in `trianglePAB,`
`angleAPB=180^(@)-(45^(@)+45^(@))" "`(angle sum property)
`=90^(@)`
In right `trianglePAB,` by Pythagoras theorem,
`AB^(2)=PA^(2)+PB^(2)=(4)^(2)+(4)^(2)=32`
`:." "AB=sqrt(32)cm=4sqrt(2)cm`
Hence, the length of chord AB is `4sqrt(2)cm.`