Question

Seg AB is a diameter of a circle with cente P , seg AC is a chord . A through P and parallel to seg AC intersects the tangent drawn at C is D . Prove that line DB is a tangent to the circle .

Answer 1

In `Delta PAC` ,
seg `PA cong seg PC " "` …(Radii of the same circle )
` :. Angle PAC cong angle PCA " "` …(Isosceles Delta theorem ) …(1)
seg AC `||` seg PD nd PA is the transversal ,
` angle PAC cong angle BPD " " ` ...(Corresponding angles ) ... (2)
seg AC `||` seg PD and PC is the transversal ,
` angle PCA cong angle CPD " "` ...(ALternate angles ) ...(3)
` :. angle BPD cong angle CPD " "` ... [ From (1) , (2) and (3) ] ...(4)
In `Delta BPD and Delta CPD,`
seg `BP cong seg CP" "` ...(Radii of the same circle )
` angle BPD cong angle CPD " " ` ...[ From (4)]
seg `PD cong seg PD " "` ...(Common side )
` :. Delta BPD cong Delta CPD " "` (SAS test)
` :. angle PBD cong angle PCD " " ` ... (c.a.c.t.)
` angle PCD = 90^(@) " "` ... (Tangent is perpendicular to radius )
` :. angle PBD = 90^(@)`
` :. ` line DB is tangent to the circle at point B
... ( Converse of tangent theorem )