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. - Mathematics

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Sum

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.

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Solution


Let us draw a circle in which AMB is an arc and M is the mid-point of the arc AMB.

Joined AM and MB.

Also TT' is a tangent at point M on the circle.

To Prove: AB || TT'

Proof: As M is the mid point of Arc AMB

Arc AM = Arc MB

AM = MB  ...[As equal chords cuts equal arcs]

∠ABM = ∠BAM  ...[Angles opposite to equal sides are equal] [1]

Now, ∠BMT' = ∠BAM  ...[Angle between tangent and the chord equals angle made by the chord in alternate segment] [2]

From [1] and [2]

∠ABM = ∠BMT'

So, AB || TT'   ...[Two lines are parallel if the interior alternate angles are equal]

Hence Proved!

  Is there an error in this question or solution?
Chapter 9: Circles - Exercise 9.4 [Page 111]

APPEARS IN

NCERT Exemplar Mathematics Class 10
Chapter 9 Circles
Exercise 9.4 | Q 9 | Page 111
RD Sharma Class 10 Maths
Chapter 8 Circles
Exercise 8.2 | Q 16 | Page 35

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