`PQ` is a chord of length `4.8cm` of a circle of radius `3cm`. The tangents at `P` and `Q` intersect at a point `T` as shown in the figure. Find the length of `TP`.

Explanation

`R` is the midpoint of `PQ`, i.e., `PR=QR=((1)/(2)xx4.8)cm=2.4cm`.
Also, `TObotPQ`.
In right `DeltaPRO`, we have
`PO^(2)=PR^(2)+RO^(2)impliesRO=sqrt(PO^(2)-PR^(2))`
`=sqrt(3^(2)-2.4^(2))=sqrt(3.24)=1.8`.
Let `TR=x` and `TP=y`.
In right `DeltaPTR`, we have `TP^(2)=TR^(2)+PR^(2)impliesy^(2)=x^(2)+(2.4)^(2)`.........`(i)`
In right `DeltaOTP`, we have
`TO^(2)=TP^(2)+PO^(2)implies(TR+RO)^(2)=TP^(2)+PO^(2)`
`implies(x+1.8)^(2)=y^(2)+3^(2)impliesx^(2)+3.6x+3.24=y^(2)+9`. ............`(ii)`
Solving `(i)` and `(ii)` , we get `x=3.2`, `y=4`.
`:. TP=4cm`.