Three points P,Q ,R lie on a circle. The two perpendiculars PQ and PR and the point P intersect the circle at the points S and T respectively. Prove that `RQ=ST`

Answer

page no 291

Explanation

Three point P,Q,R are on the circle with centre at O.PS ` bot PQ and PT bot PR`
To prove `RQ= ST`.
Constraction : Let us join the points `Q,S,R,T,P,S and S, T`
Proof : `PS bot PQ :. angle QPS=90^(@)`
`:. angle QPS` is a semi-circular angle.

`:.QS` is a diameter, of which O is the mid-point `:. OQ=OS=` radius Similarly, `PT bot PR. :. angle RPT=90^(@)`
`:. angle RPT` is semic-circular angle.
`:.RT` is a diameterof which O is the mid-point
`:. OR=OT` [ for similar reason]and `angleQOR= angle SOT [:.` oppoiste circle]
`:. Delta OQR~=Delta OST` [ by the S-A-S condition of congruency]
`:. QR= ST[ :.` similar sides of congruent triangle]
Hence RQ= ST.