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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:
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