PR is a diameter of a circle. A tangent is draewn at the point P and a point S is taken on the tangent of the circle is such a way that PR=PS. If RS intersects the circle at the point T., prove that ST =PT.

Explanation

Let PR be a diameter of the circle with centre at O. A tangent AB is draen at P. The part PS from AB is cut equal to PR. RS intersects the circle at the point T.
To prove : `ST=PT`
Proof : `because AB` is a tangent at P on the circle with centre at O and OP is a radius passing through `P. therefore OP bot AB.`
`therefore angle OPS =90^(@)or, angle RPT+angle TPS=90^(@).....(1)`
Again, `angle PTR` is a semicircular angle,` therefore angle PTR=90^(@)`
`therefore angle TPR+angle TRP =90^(@).....(2)`
From (1) and (2) we get, `angle RPT +angle TPS =angle TPS=angle TPR +angle TRP`
or, `angle TPS =angle TRP because angle RPT =angle TPR or, angle TPS =angle TSP[because PS = RP,`
`therefoer angle TRP =angle RSP]`
`therefore ST= PT[because` opposite side of equal angles ar equal.]
Hence `ST=PT` (Proved)