Math, asked by ankitasamantaray292, 7 months ago

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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Answers

Answered by devirekha7559
7

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.

... ✌✌MARK AS BRAINLIEST..

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