Prove that tangent drawn at any point of a circle is perpendicular to the
radius through the point of contact.
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ɢɪᴠᴇɴ : ᴀ ᴄɪʀᴄʟᴇ ᴄ (0, ʀ) ᴀɴᴅ ᴀ ᴛᴀɴɢᴇɴᴛ ʟ ᴀᴛ ᴘᴏɪɴᴛ ᴀ.
ᴛᴏ ᴘʀᴏᴠᴇ : ᴏᴀ ⊥ ʟ
ᴄᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ : ᴛᴀᴋᴇ ᴀ ᴘᴏɪɴᴛ ʙ, ᴏᴛʜᴇʀ ᴛʜᴀɴ ᴀ, ᴏɴ ᴛʜᴇ ᴛᴀɴɢᴇɴᴛ ʟ. ᴊᴏɪɴ ᴏʙ. ꜱᴜᴘᴘᴏꜱᴇ ᴏʙ ᴍᴇᴇᴛꜱ ᴛʜᴇ ᴄɪʀᴄʟᴇ ɪɴ ᴄ.
ᴘʀᴏᴏꜰ: ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ, ᴀᴍᴏɴɢ ᴀʟʟ ʟɪɴᴇ ꜱᴇɢᴍᴇɴᴛ ᴊᴏɪɴɪɴɢ ᴛʜᴇ ᴘᴏɪɴᴛ ᴏ ᴛᴏ ᴀ ᴘᴏɪɴᴛ ᴏɴ ʟ, ᴛʜᴇ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ɪꜱ ꜱʜᴏʀᴛᴇꜱᴛ ᴛᴏ ʟ.
ᴏᴀ = ᴏᴄ (ʀᴀᴅɪᴜꜱ ᴏꜰ ᴛʜᴇ ꜱᴀᴍᴇ ᴄɪʀᴄʟᴇ)
ɴᴏᴡ, ᴏʙ = ᴏᴄ + ʙᴄ.
∴ ᴏʙ > ᴏᴄ
⇒ ᴏʙ > ᴏᴀ
⇒ ᴏᴀ < ᴏʙ
ʙ ɪꜱ ᴀɴ ᴀʀʙɪᴛʀᴀʀʏ ᴘᴏɪɴᴛ ᴏɴ ᴛʜᴇ ᴛᴀɴɢᴇɴᴛ ʟ. ᴛʜᴜꜱ, ᴏᴀ ɪꜱ ꜱʜᴏʀᴛᴇʀ ᴛʜᴀɴ ᴀɴʏ ᴏᴛʜᴇʀ ʟɪɴᴇ ꜱᴇɢᴍᴇɴᴛ ᴊᴏɪɴɪɴɢ ᴏ ᴛᴏ ᴀɴʏ ᴘᴏɪɴᴛ ᴏɴ ʟ.
ʜᴇʀᴇ, ᴏᴀ ⊥ ʟ
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