In the given figure, XY and X'Y' are two
parallel tangents to a circle with centre O and
another tangent AB with point of contact C
intersecting XY at A and X'Y' at B. Prove that
<AOB = 90
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ɪɴ △ᴀᴏᴘ & △ᴀᴏᴄ
ᴏᴘ=ᴏᴄ [ʙᴏᴛʜ ʀᴀᴅɪᴜs]
ᴀᴘ=ᴀᴄ [ʟᴇɴɢᴛʜ ᴏғ ᴛᴀɴɢᴇɴᴛsᴅʀᴀᴡɴ ғʀᴏᴍ ᴇxᴛᴇʀɴᴀʟ ᴘᴏɪɴᴛ ᴛᴏ ᴀ ᴄɪʀᴄʟᴇ ᴀʀᴇ ᴇǫᴜᴀʟ]
ᴏᴀ=ᴏᴀ [ᴄᴏᴍᴍᴏɴ]
∴△ᴀᴏᴄ≅△ᴀᴏᴘ [sss ᴄᴏɴɢʀᴜᴇɴᴄᴇ ʀᴜʟᴇ]
sᴏ, ∠ᴀᴏᴘ=∠ᴀᴏᴄ...............(1) [ᴄᴘᴄᴛ]
ɴᴏᴡ,
ɪɴ △ʙᴏᴄ & △ʙᴏǫ
ᴏᴄ=ᴏǫ [ʙᴏᴛʜ ʀᴀᴅɪᴜs]
ʙᴄ=ʙǫ [ʟᴇɴɢᴛʜ ᴏғ ᴛᴀɴɢᴇɴᴛs ᴅʀᴀᴡɴ ғʀᴏᴍ ᴇxᴛᴇʀɴᴀʟ ᴘᴏɪɴᴛ ᴛᴏ ᴀ ᴄɪʀᴄʟᴇ ᴀʀᴇ ᴇǫᴜᴀʟ]
ᴏʙ=ᴏʙ [ᴄᴏᴍᴍᴏɴ]
∴△ʙᴏᴄ≅△ʙᴏǫ [sss ᴄᴏɴɢʀᴜᴇɴᴄᴇ ʀᴜʟᴇ]
sᴏ, ∠ʙᴏᴄ=∠ʙᴏǫ...............(2) [ᴄᴘᴄᴛ]
ғᴏʀ ʟɪɴᴇ ᴘǫ:
∠ᴀᴏᴘ+∠ᴀᴏᴄ+∠ʙᴏᴄ+∠ʙᴏǫ=180
∠ᴀᴏᴄ+∠ᴀᴏᴄ+∠ʙᴏᴄ+∠ʙᴏᴄ=180
2∠ᴀᴏᴄ+2∠ʙᴏᴄ=180
2(∠ᴀᴏᴄ+∠ʙᴏᴄ)=180
∠ᴀᴏᴄ+∠ʙᴏᴄ= 180/2
∠ᴀᴏᴄ+∠ʙᴏᴄ=90
∠ᴀᴏʙ=90
ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ.
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