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  • 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.

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

Answers (1)

Solution
           

Let mid-point of arc is C and DCE be the tangent to the circle.
  Construction: Join AB, AC and BC.
Proof: In \bigtriangleupABC
AC = BC
   \Rightarrow \, \angle CAB= \angle CBA\: \cdots \left ( i \right )                   
[  sides opposite to equal angles are equal]
Here DCF is a tangent line
\therefore \; \; \angle ACD= \angle CBA [  angle in alternate segments are equal]
   \Rightarrow \; \angle ACD= \angle CAB …..(ii)         [From equation (i)]
But Here \angle ACD  and \angle CAB  are alternate angels.
\therefore equation (ii) holds only when AB||DCE.

Hence 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.

Hence Proved.

 

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