Question

In the given figure, AB and CD are two chords of a circle with centre O such that MP=NP.
If OM is perpendicular to AB and ON is perpendicular to DC, show that AB = CD.​

Answer 1

Refer to the solution in the image and use the theorem that Perpendicular drawn from center bisects the chord.

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Answer 2

$\huge\fcolorbox{black}{aqua}{Solution:-}$

✒ ɢɪᴠᴇɴ:-

⛄ ᴀʙ ᴀɴᴅ ᴄᴅ ᴀʀᴇ ᴛᴡᴏ ᴄʜᴏʀᴅs ᴏғ ᴀ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛʀᴇ ᴏ.

⛄ ᴍᴘ=ɴᴘ.

⛄ "ᴏᴍ" ɪs ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴛᴏ "ᴀʙ" ᴀɴᴅ "ᴏɴ" ɪs ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴛᴏ "ᴅᴄ".

✒ ᴛᴏ ᴘʀᴏᴠᴇ:-

⛄ᴀʙ = ᴄᴅ.

✒ ᴘʀᴏᴏғ:-

⛄ ᴄᴏɴsɪᴅᴇʀ ∆ᴏᴍᴘ ᴀɴᴅ ∆ᴏɴᴘ. ʜᴇʀᴇ,

↪ᴀɴɢʟᴇ ᴏᴍᴘ = ᴀɴɢʟᴇ ᴏɴᴘ = 90° [ɢɪᴠᴇɴ]

↪ᴏᴘ = ᴏᴘ [ᴄᴏᴍᴍᴏɴ]

↪ᴍᴘ = ɴᴘ [ɢɪᴠᴇɴ]

ᴛʜᴇʀᴇғᴏʀᴇ,

∆ᴏᴍᴘ ≈ ∆ᴏɴᴘ [ʙʏ ʀʜs ʀᴜʟᴇ]

=) ᴏᴍ = ᴏɴ [ʙʏ ᴄᴘᴄᴛ]

=) ᴀʙ = ᴄᴅ

[sɪɴᴄᴇ ᴄʜᴏʀᴅs ᴇϙᴜɪᴅɪsᴛᴀɴᴛ ғʀᴏᴍ ᴛʜᴇ ᴄᴇɴᴛʀᴇ ᴀʀᴇ ᴇϙᴜᴀʟ]

Ⓗⓔⓝⓒⓔ Ⓟⓡⓞⓥⓔⓓ!!

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