EXAMPLE 11Prove that the tangent drawn at the midpoint of an arc of a

1.	EXAMPLE 11Prove that the tangent drawn at the midpoint of an arc of a circle isparallel to the chord joining the end points of the arc.ICBSE 2015]SOLUTIONGVEN Point Pis the midpoint of arc QR of a circle with centreO. ST is the tangent to the circle at point P.TO PROVE Chord QRI\|ST.PROOF P is the midpoint of QR0.chord QP chord PR
Answer» A circle with Centre O, P is the midpoint of Arc APB. PT is a tangent to the circle at P. To Prove: AB \|\| PT Construction: join OA ,OB, & OP Proof: OP ⟂PT [Radius is ⟂ to tangent through point of contact] ∠OPT= 90° Since P is the midpoint of Arc APB Arc AAP =arc BP ∠AOP = ∠BOP ∠AOM= ∠BOM In ∆ AOM & ∆BOM OA= OB= r OM = OM (Common) ∠AOM= ∠BOM (proved above) ∠AOM≅∠BOM (by SAS congruency axiom) ∠AMO = ∠BMO (c.p.c.t) ∠AMO + ∠BMO= 180° ∠AMO = ∠BMO= 90° ∠BMO = ∠OPT= 90° But, they are corresponding angles. Hence, AD\|\|PT

EXAMPLE 11Prove that the tangent drawn at the midpoint of an arc of a circle isparallel to the chord joining the end points of the arc.ICBSE 2015]SOLUTIONGVEN Point Pis the midpoint of arc QR of a circle with centreO. ST is the tangent to the circle at point P.TO PROVE Chord QRI|ST.PROOF P is the midpoint of QR0.chord QP chord PR

Answer»

A circle with Centre O, P is the midpoint of Arc APB. PT is a tangent to the circle at P.

To Prove:

AB || PT

Construction: join OA ,OB, & OP

Proof: OP ⟂PT

[Radius is ⟂ to tangent through point of contact]

∠OPT= 90°

Since P is the midpoint of Arc APB

Arc AAP =arc BP

∠AOP = ∠BOP

∠AOM= ∠BOM

In ∆ AOM & ∆BOM

OA= OB= r

OM = OM (Common)

∠AOM= ∠BOM (proved above)

∠AOM≅∠BOM (by SAS congruency axiom)

∠AMO = ∠BMO (c.p.c.t)

∠AMO + ∠BMO= 180°

∠AMO = ∠BMO= 90°

∠BMO = ∠OPT= 90°

But, they are corresponding angles. Hence, AD||PT