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Answer 1

AnSwEr:

ɢɪᴠᴇɴ : ғʀᴏᴍ ᴀɴ ᴇxᴛᴇʀɴᴀʟ ᴘᴏɪɴᴛ ᴘ, ᴛᴡᴏ ᴛᴀɴɢᴇɴᴛs ᴘᴛ ᴀɴᴅ ᴘs ᴀʀᴇ ᴅʀᴀᴡɴ ᴛᴏ ᴀ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛᴇʀ ᴏ ᴀɴᴅ ʀᴀᴅɪᴜs ʀ ᴀɴᴅ ᴏᴘ = 2ʀ

ᴛᴏ ᴘʀᴏᴠᴇ : ∠ᴏᴛs = ∠ᴏsᴛ = 30°

ᴘʀᴏᴏғ :

ᴏs = ᴏᴛ [ʀᴀᴅɪɪ ᴏғ sᴀᴍᴇ ᴄɪʀᴄʟᴇ]

∠ᴏᴛs = ∠ᴏsᴛ [ᴀɴɢʟᴇs ᴏᴘᴘᴏsɪᴛᴇ ᴛᴏ ᴇϙᴜᴀʟ sɪᴅᴇs ᴀʀᴇ ᴇϙᴜᴀʟ] ….[1]

ᴀs ᴛᴀɴɢᴇɴᴛ ᴅʀᴀᴡɴ ᴀᴛ ᴀ ᴘᴏɪɴᴛ ᴏɴ ᴄɪʀᴄʟᴇ ɪs ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴛᴏ ᴛʜᴇ ʀᴀᴅɪᴜs ᴛʜʀᴏᴜɢʜ ᴘᴏɪɴᴛ ᴏғ ᴄᴏɴᴛᴀᴄᴛ.

ᴏᴛ⏊ᴛᴘ ᴀɴᴅ ᴏs⏊sᴘ

∠ᴏsᴘ = 90°

∠ᴏsᴛ + ∠ᴘsᴛ = 90°

∠ᴘsᴛ = 90° - ∠ᴏsᴛ …[2]

ɪɴ ᴛʀɪᴀɴɢʟᴇ ᴘᴛs

ᴘᴛ = ᴘs [ ᴛᴀɴɢᴇɴᴛs ᴅʀᴀᴡɴ ғʀᴏᴍ ᴀɴ ᴇxᴛᴇʀɴᴀʟ ᴘᴏɪɴᴛ ᴛᴏ ᴀ ᴄɪʀᴄʟᴇ ᴀʀᴇ ᴇϙᴜᴀʟ]

∠ᴘsᴛ = ∠ᴘᴛs = 90° - ∠ᴏsᴛ [ғʀᴏᴍ 2]

ɪɴ △ᴘᴛs

∠ᴘᴛs + ∠ᴘsᴛ + ∠sᴘᴛ = 180° [ᴀɴɢʟᴇ sᴜᴍ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴀ ᴛʀɪᴀɴɢʟᴇ]

90 - ∠ᴏsᴛ + 90 - ∠ᴏsᴛ + ∠sᴘᴛ = 180

∠sᴘᴛ = 2∠ᴏsᴛ …[3]

ɪɴ △ᴏᴛᴘ, ᴏᴛ⏊ᴛᴘ

∠ᴏᴘᴛ = 30° …[4]

sɪᴍɪʟᴀʀʟʏ, ɪɴ △ᴏsᴘ

∠ᴏᴘs = 30° …[5]

ᴀᴅᴅɪɴɢ [4] ᴀɴᴅ [5]

∠ᴏᴘᴛ + ∠ᴏᴘs = 30° + 30°

∠sᴘᴛ = 60°

ɴᴏᴡ ᴘᴜᴛᴛɪɴɢ ᴛʜɪs ᴠᴀʟᴜᴇ ɪɴ [3]

∠sᴘᴛ = 2∠ᴏsᴛ

60 = 2∠ᴏsᴛ

∠ᴏsᴛ = 30° …[6]

ғʀᴏᴍ [1] ᴀɴᴅ [6]

∠ᴏsᴛ = ∠ᴏᴛs = 30°

Answer 2

Answer:

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