Question

prove that of the equilateral triangle described on the side of a square is half of the area of the equilateral triangle described on it's diagonal​

Answer 1

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⎟⎟ ✪✪ ＱＵＥＳＴＩＯＮ ✪✪⎟⎟

ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ᴏғ ᴛʜᴇ ᴇϙᴜɪʟᴀᴛᴇʀᴀʟ ᴛʀɪᴀɴɢʟᴇ ᴅᴇsᴄʀɪʙᴇᴅ ᴏɴ ᴛʜᴇ sɪᴅᴇ ᴏғ ᴀ sϙᴜᴀʀᴇ ɪs ʜᴀʟғ ᴏғ ᴛʜᴇ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ ᴇϙᴜɪʟᴀᴛᴇʀᴀʟ ᴛʀɪᴀɴɢʟᴇ ᴅᴇsᴄʀɪʙᴇᴅ ᴏɴ ɪᴛ's ᴅɪᴀɢᴏɴᴀʟ

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⎟⎟ ✿✿ ＡＮＳＷＥＲ ✿✿ ⎟⎟

Bʏ ᴛʜᴇ ɢɪᴠᴇɴ ғɪɢᴜʀᴇ

AB = AC = BE

AC = AF = CF

Bʏ Pʏᴛʜᴀɢᴏʀᴀs ᴛʜᴇᴏʀᴇᴍ :-

ɪɴ ∆ᴀʙᴄ = ᴀʙ^2 + ʙᴄ^2 = ᴀᴄ^2

➠ ａ^2 + ａ^2 = ＡＣ^2

➠ AC^2 = 2a^2

.°. ᴀᴄ = √2 a

ᴀʀᴇᴀ ᴏғ ∆ᴀʙᴇ = √3/4 × a^2

ᴀʀᴇᴀ ᴏғ ∆ᴀᴄғ = √3/4 (√2a ) ^2 ➠ 2 √3/4

.°. ᴀʀᴇᴀ ᴏғ ∆ᴀʙᴇ / ᴀʀᴇᴀ ᴏғ ∆ᴀᴄғ = 1 / 2

Hᴇɴᴄᴇ Pʀᴏᴠᴇᴅ

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Answer 2

plz refer to this attachment