Question

PQ is chord parallel to a tangent at the circle at R. prove that R bisects the arc PRQ

Answer 1

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$\large \underline \red{ᴀɴsᴡᴇʀ: - }$

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQ

ᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2

⇨MRN∣∣PQ

∴ ∠1=∠3 [Alternate interior angles]

⇨∠2=∠3

⇨PR=RQ [Sides opp. to equal ∠s in ΔRPQ]

∵ Equal chords subtend equal arcs in a circle so

arcPR=arc RQ or R bisect the arc PRQ.

• Hence proved.

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Answer 2

Answer:

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQ

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ∴ ∠1=∠3 [Alternate interior angles]

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ∴ ∠1=∠3 [Alternate interior angles]⇨∠2=∠3

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ∴ ∠1=∠3 [Alternate interior angles]⇨∠2=∠3⇨PR=RQ [Sides opp. to equal ∠s in ΔRPQ]

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ∴ ∠1=∠3 [Alternate interior angles]⇨∠2=∠3⇨PR=RQ [Sides opp. to equal ∠s in ΔRPQ]∵ Equal chords subtend equal arcs in a circle so

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ∴ ∠1=∠3 [Alternate interior angles]⇨∠2=∠3⇨PR=RQ [Sides opp. to equal ∠s in ΔRPQ]∵ Equal chords subtend equal arcs in a circle soarcPR=arc RQ or R bisect the arc PRQ.

ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : - Join RP and RQᴘʀᴏᴏꜰ :- Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2⇨MRN∣∣PQ∴ ∠1=∠3 [Alternate interior angles]⇨∠2=∠3⇨PR=RQ [Sides opp. to equal ∠s in ΔRPQ]∵ Equal chords subtend equal arcs in a circle soarcPR=arc RQ or R bisect the arc PRQ.Hence proved.