Question

write the besic proportionality thorem​

Answer 1

Answer:

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Step-by-step explanation:

Basic Proportionality Theorem states that "If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio".

Answer 2

ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ sᴛᴀᴛᴇs ᴛʜᴀᴛ "ɪғ ᴀ ʟɪɴᴇ ɪs ᴅʀᴀᴡɴ ᴘᴀʀᴀʟʟᴇʟ ᴛᴏ ᴏɴᴇ sɪᴅᴇ ᴏғ ᴀ ᴛʀɪᴀɴɢʟᴇ ᴛᴏ ɪɴᴛᴇʀsᴇᴄᴛ ᴛʜᴇ ᴏᴛʜᴇʀ ᴛᴡᴏ sɪᴅᴇs ɪɴ ᴅɪsᴛɪɴᴄᴛ ᴘᴏɪɴᴛs, ᴛʜᴇ ᴏᴛʜᴇʀ ᴛᴡᴏ sɪᴅᴇs ᴀʀᴇ ᴅɪᴠɪᴅᴇᴅ ɪɴ ᴛʜᴇ sᴀᴍᴇ ʀᴀᴛɪᴏ".

ᴄᴏɴsɪᴅᴇʀ ᴀ ᴛʀɪᴀɴɢʟᴇ Δᴀʙᴄ,ɪɴ ᴛʜɪs ᴛʀɪᴀɴɢʟᴇ, ᴡᴇ ᴅʀᴀᴡ ᴀ ʟɪɴᴇ ᴘǫ ᴘᴀʀᴀʟʟᴇʟ ᴛᴏ ᴛʜᴇ sɪᴅᴇ ʙᴄ ᴏғ Δᴀʙᴄ ᴀɴᴅ ɪɴᴛᴇʀsᴇᴄᴛɪɴɢ ᴛʜᴇ sɪᴅᴇs ᴀʙ ᴀɴᴅ ᴀᴄ ɪɴ ᴘ ᴀɴᴅ ǫ, ʀᴇsᴘᴇᴄᴛɪᴠᴇʟʏ

ᴊᴏɪɴ ᴛʜᴇ ᴠᴇʀᴛᴇx ʙ ᴏғ Δᴀʙᴄ ᴛᴏ ǫ ᴀɴᴅ ᴛʜᴇ ᴠᴇʀᴛᴇx ᴄ ᴛᴏ ᴘ ᴛᴏ ғᴏʀᴍ ᴛʜᴇ ʟɪɴᴇs ʙǫ ᴀɴᴅ ᴄᴘ ᴀɴᴅ ᴛʜᴇɴ ᴅʀᴏᴘ ᴀ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ǫɴ ᴛᴏ ᴛʜᴇ sɪᴅᴇ ᴀʙ ᴀɴᴅ ᴀʟsᴏ ᴅʀᴀᴡ ᴘᴍ⊥ᴀᴄ

ɴᴏᴡ ᴛʜᴇ ᴀʀᴇᴀ ᴏғ ∆ᴀᴘǫ = 1/2 × ᴀᴘ × ǫɴ (sɪɴᴄᴇ, ᴀʀᴇᴀ ᴏғ ᴀ ᴛʀɪᴀɴɢʟᴇ= 1/2× ʙᴀsᴇ × ʜᴇɪɢʜᴛ)

sɪᴍɪʟᴀʀʟʏ, ᴀʀᴇᴀ ᴏғ ∆ᴘʙǫ= 1/2 × ᴘʙ × ǫɴ

ᴀʀᴇᴀ ᴏғ ∆ᴀᴘǫ = 1/2 × ᴀǫ × ᴘᴍ

ᴀʟsᴏ,ᴀʀᴇᴀ ᴏғ ∆ǫᴄᴘ = 1/2 × ǫᴄ × ᴘᴍ ………… (1)

ɴᴏᴡ, ɪғ ᴡᴇ ғɪɴᴅ ᴛʜᴇ ʀᴀᴛɪᴏ ᴏғ ᴛʜᴇ ᴀʀᴇᴀ ᴏғ ᴛʀɪᴀɴɢʟᴇs ∆ᴀᴘǫᴀɴᴅ ∆ᴘʙǫ, ᴡᴇ ʜᴀᴠᴇ

ᴀʀᴇᴀ ᴏғ Δᴀᴘǫᴀʀᴇᴀ ᴏғ Δᴘʙǫ = 12 × ᴀᴘ × ǫɴ12 × ᴘʙ × ǫɴ = ᴀᴘᴘʙ

sɪᴍɪʟᴀʀʟʏ, ᴀʀᴇᴀ ᴏғ Δᴀᴘǫᴀʀᴇᴀ ᴏғ Δǫᴄᴘ = 12 × ᴀǫ × ᴘᴍ12 × ǫᴄ × ᴘᴍ = ᴀǫǫᴄ ………..(2)

ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ᴛʜᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴛʀɪᴀɴɢʟᴇs, ᴛʜᴇ ᴛʀɪᴀɴɢʟᴇs ᴅʀᴀᴡɴ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ sᴀᴍᴇ ᴘᴀʀᴀʟʟᴇʟ ʟɪɴᴇs ᴀɴᴅ ᴏɴ ᴛʜᴇ sᴀᴍᴇ ʙᴀsᴇ ʜᴀᴠᴇ ᴇǫᴜᴀʟ ᴀʀᴇᴀs.

ᴛʜᴇʀᴇғᴏʀᴇ, ᴡᴇ ᴄᴀɴ sᴀʏ ᴛʜᴀᴛ ∆ᴘʙǫ ᴀɴᴅ ǫᴄᴘ ʜᴀᴠᴇ ᴛʜᴇ sᴀᴍᴇ ᴀʀᴇᴀ.

ᴀʀᴇᴀ ᴏғ ∆ᴘʙǫ = ᴀʀᴇᴀ ᴏғ ∆ǫᴄᴘ …………..(3)

ᴛʜᴇʀᴇғᴏʀᴇ, ғʀᴏᴍ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴs (1), (2) ᴀɴᴅ (3) ᴡᴇ ᴄᴀɴ sᴀʏ ᴛʜᴀᴛ,

ᴀᴘ/ᴘʙ = ᴀǫ/ǫᴄ

ᴀʟsᴏ, ∆ᴀʙᴄ ᴀɴᴅ ∆ᴀᴘǫ ғᴜʟғɪʟ ᴛʜᴇ ᴄᴏɴᴅɪᴛɪᴏɴs ғᴏʀ sɪᴍɪʟᴀʀ ᴛʀɪᴀɴɢʟᴇs, ᴀs sᴛᴀᴛᴇᴅ ᴀʙᴏᴠᴇ. ᴛʜᴜs, ᴡᴇ ᴄᴀɴ sᴀʏ ᴛʜᴀᴛ ∆ᴀʙᴄ ~∆ᴀᴘǫ.

ᴛʜᴇ ᴍɪᴅᴘᴏɪɴᴛ ᴛʜᴇᴏʀᴇᴍ ɪs ᴀ sᴘᴇᴄɪᴀʟ ᴄᴀsᴇ ᴏғ ᴛʜᴇ ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ.

ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ᴍɪᴅ-ᴘᴏɪɴᴛ ᴛʜᴇᴏʀᴇᴍ, ᴀ ʟɪɴᴇ ᴅʀᴀᴡɴ ᴊᴏɪɴɪɴɢ ᴛʜᴇ ᴍɪᴅᴘᴏɪɴᴛs ᴏғ ᴛʜᴇ ᴛᴡᴏ sɪᴅᴇs ᴏғ ᴀ ᴛʀɪᴀɴɢʟᴇ ɪs ᴘᴀʀᴀʟʟᴇʟ ᴛᴏ ᴛʜᴇ ᴛʜɪʀᴅ sɪᴅᴇ.

ᴄᴏɴsɪᴅᴇʀ ᴀɴ ∆ᴀʙᴄ.