Question

Derive the Pythagoras Theorem. ​

Answer 1

ᴄᴏɴsɪᴅᴇʀ ᴀ ʀɪɢʜᴛ-ᴀɴɢʟᴇᴅ ᴛʀɪᴀɴɢʟᴇ Δᴀʙᴄ. ғʀᴏᴍ ᴛʜᴇ ʙᴇʟᴏᴡ ғɪɢᴜʀᴇ, ɪᴛ ɪs ʀɪɢʜᴛ-ᴀɴɢʟᴇᴅ ᴀᴛ ʙ.

ᴘʏᴛʜᴀɢᴏʀᴀs ᴛʜᴇᴏʀᴇᴍ ᴅᴇʀɪᴠᴀᴛɪᴏɴ -1

ʟᴇᴛ ʙᴅ ʙᴇ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴛᴏ ᴛʜᴇ sɪᴅᴇ ᴀᴄ.

ᴘʏᴛʜᴀɢᴏʀᴀs ᴛʜᴇᴏʀᴇᴍ ᴅᴇʀɪᴠᴀᴛɪᴏɴ -2

ғʀᴏᴍ ᴛʜᴇ ᴀʙᴏᴠᴇ-ɢɪᴠᴇɴ ғɪɢᴜʀᴇ, ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ Δᴀʙᴄ ᴀɴᴅ Δᴀᴅʙ,

ɪɴ Δᴀʙᴄ ᴀɴᴅ Δᴀᴅʙ,

∠ᴀʙᴄ = ∠ᴀᴅʙ = 90°

∠ᴀ = ∠ᴀ → ᴄᴏᴍᴍᴏɴ

ᴜsɪɴɢ ᴛʜᴇ ᴀᴀ ᴄʀɪᴛᴇʀɪᴏɴ ғᴏʀ ᴛʜᴇ sɪᴍɪʟᴀʀɪᴛʏ ᴏғ ᴛʀɪᴀɴɢʟᴇs,

Δᴀʙᴄ ~ Δᴀᴅʙ

ᴛʜᴇʀᴇғᴏʀᴇ, ᴀᴅ/ᴀʙ = ᴀʙ/ᴀᴄ

⇒ ᴀʙ2 = ᴀᴄ x ᴀᴅ ……(1)

ᴄᴏɴsɪᴅᴇʀɪɴɢ Δᴀʙᴄ ᴀɴᴅ Δʙᴅᴄ ғʀᴏᴍ ᴛʜᴇ ʙᴇʟᴏᴡ ғɪɢᴜʀᴇ.

ᴘʏᴛʜᴀɢᴏʀᴀs ᴛʜᴇᴏʀᴇᴍ ᴅᴇʀɪᴠᴀᴛɪᴏɴ -3

∠ᴄ = ∠ᴄ → ᴄᴏᴍᴍᴏɴ

∠ᴄᴅʙ = ∠ᴀʙᴄ = 90°

ᴜsɪɴɢ ᴛʜᴇ ᴀɴɢʟᴇ ᴀɴɢʟᴇ(ᴀᴀ) ᴄʀɪᴛᴇʀɪᴏɴ ғᴏʀ ᴛʜᴇ sɪᴍɪʟᴀʀɪᴛʏ ᴏғ ᴛʀɪᴀɴɢʟᴇs, ᴡᴇ ᴄᴏɴᴄʟᴜᴅᴇ ᴛʜᴀᴛ,

Δʙᴅᴄ ~ Δᴀʙᴄ

ᴛʜᴇʀᴇғᴏʀᴇ, ᴄᴅ/ʙᴄ = ʙᴄ/ᴀᴄ

⇒ ʙᴄ2 = ᴀᴄ x ᴄᴅ …..(2)

ғʀᴏᴍ ᴛʜᴇ sɪᴍɪʟᴀʀɪᴛʏ ᴏғ ᴛʀɪᴀɴɢʟᴇs, ᴡᴇ ᴄᴏɴᴄʟᴜᴅᴇ ᴛʜᴀᴛ,

∠ᴀᴅʙ = ∠ᴄᴅʙ = 90°

sᴏ ɪғ ᴀ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ɪs ᴅʀᴀᴡɴ ғʀᴏᴍ ᴛʜᴇ ʀɪɢʜᴛ-ᴀɴɢʟᴇᴅ ᴠᴇʀᴛᴇx ᴏғ ᴀ ʀɪɢʜᴛ ᴛʀɪᴀɴɢʟᴇ ᴛᴏ ᴛʜᴇ ʜʏᴘᴏᴛᴇɴᴜsᴇ, ᴛʜᴇɴ ᴛʜᴇ ᴛʀɪᴀɴɢʟᴇs ғᴏʀᴍᴇᴅ ᴏɴ ʙᴏᴛʜ sɪᴅᴇs ᴏғ ᴛʜᴇ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴀʀᴇ sɪᴍɪʟᴀʀ ᴛᴏ ᴇᴀᴄʜ ᴏᴛʜᴇʀ ᴀɴᴅ ᴀʟsᴏ ᴛᴏ ᴛʜᴇ ᴡʜᴏʟᴇ ᴛʀɪᴀɴɢʟᴇ.

ᴛᴏ ᴘʀᴏᴠᴇ: ᴀᴄ2 =ᴀʙ2 +ʙᴄ2

ʙʏ ᴀᴅᴅɪɴɢ ᴇǫᴜᴀᴛɪᴏɴ (1) ᴀɴᴅ ᴇǫᴜᴀᴛɪᴏɴ (2), ᴡᴇ ɢᴇᴛ:

ᴀʙ2 + ʙᴄ2= (ᴀᴄ x ᴀᴅ) + (ᴀᴄ x ᴄᴅ)

ᴀʙ2 + ʙᴄ2 = ᴀᴄ (ᴀᴅ + ᴄᴅ) …..(3)

sɪɴᴄᴇ ᴀᴅ + ᴄᴅ = ᴀᴄ, sᴜʙsᴛɪᴛᴜᴛᴇ ᴛʜɪs ᴠᴀʟᴜᴇ ɪɴ ᴇǫᴜᴀᴛɪᴏɴ (3).

ᴀʙ2 + ʙᴄ2= ᴀᴄ (ᴀᴄ)

ɴᴏᴡ, ɪᴛ ʙᴇᴄᴏᴍᴇs

ᴀʙ2+ ʙᴄ2= ᴀᴄ2

ʜᴇɴᴄᴇ, ᴘʏᴛʜᴀɢᴏʀᴀs ᴛʜᴇᴏʀᴇᴍ ɪs ᴘʀᴏᴠᴇᴅ.

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