Question
Mon March 16, 2015 By: Satyaranjan Satapathy

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Expert Reply
Vijaykumar Wani
Tue March 17, 2015
 
In space the space figure comma space straight C space is space the space midpoint space of space the space minor space arc space PQ comma straight O space is space the space centre space of space the space circle space and space AB space is space tangent space to space the space circle space through space point space straight C.  We space space have space to space show space the space tangent space drawn space at space the space midpoint space of space the space ars space PQ space of space straight a space circle space is space parallel space to space the space chord space joining space the space end space points space of space the space arc space PQ. straight i. straight e. space We space will space show space PQ space vertical line vertical line space AB.  It space is space gien space that space straight C space is space midpoint space of space the space arc space PQ. So comma space arc space PC space equals space arc space CQ rightwards double arrow space PC equals CQ  This space shows space that space increment PQC space is space an space isosceles space triangle. Thus comma space the space perpendicular space bisector space of space the space side space PQ space of space increment PQC space passes space through space vertex space straight C. The space perpendicular space bisector space of space straight a space chord space passes space through space the space centre space of space the space circle. So space the space perpendicular space bisector space of space PQ space space passes space through space the space centre space straight O space of space the space circle. Thus space perpendicular space bisector space of space PQ space passes space through space the space points space straight O space and space straight C. rightwards double arrow PQ perpendicular OC AB space is space the space tangent space to space the space circle space through space the space point space straight C space on space the space circle. rightwards double arrow AB perpendicular OC  The space chord space PQ space and space the space tangent space PQ space of space the space circle space are space perpendicular space to space the space same space line space OC. therefore PQ space vertical line vertical line space AB
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