Question

Consider an equilateral triangle and
form triangle by joining the mid points
of the sides of the equilateral triangle
and continue this process for infinitely
many times. Then the common ratio of
the progression of perimeters of the
equilateral triangles is ....... (​

Answer 1

Answer:-

Aʀᴇᴀ ᴏғ ᴇϙᴜɪʟᴀᴛᴇʀᴀʟ ᴛʀɪᴀɴɢʟᴇ ᴏғ sɪᴅᴇ ᴀ ɪs 34ᴀ2

ᴛʜᴇ sɪᴅᴇ ᴏғ ᴛʜᴇ ᴇϙᴜɪʟᴀᴛᴇʀᴀʟ ᴛʀɪᴀɴɢʟᴇ ғᴏʀᴍᴇᴅ ʙʏ ᴊᴏɪɴɪɴɢ ᴛʜᴇ ᴍɪᴅᴘᴏɪɴᴛs ᴏғ ᴏʀɪɢɪɴᴀʟ ᴛʀɪᴀɴɢʟᴇ ɪs ᴀ/2

ᴛʜᴇ ᴄᴏʀʀᴇsᴘᴏɴᴅɪɴɢ ᴀʀᴇᴀ ɪs 34(2ᴀ)2

ᴀɴᴅ ғᴏʀ ᴛʜᴇ ɴᴇxᴛ ᴛʀɪᴀɴɢʟᴇ ɪɴsɪᴅᴇ ᴛʜᴇ ᴀʀᴇᴀ ɪs 34(4ᴀ)2

sᴏ ᴛʜᴇ sᴜᴍ ᴏғ ᴀʟʟ ᴛʜᴇsᴇ ᴀʀᴇᴀs ɪs ᴀ ɪɴғɪɴɪᴛᴇ ɢᴇᴏᴍᴇᴛʀɪᴄ sᴇʀɪᴇs ᴡɪᴛʜ ғɪʀsᴛ ᴛᴇʀᴍ ᴀs 34ᴀ2 ᴀɴᴅ ᴡɪᴛʜ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ 1\4

sᴜᴍ ᴏғ ɪɴғɪɴɪᴛᴇ ɢᴇᴏᴍᴇᴛʀɪᴄ sᴇʀɪᴇs ɪs 1−ᴄᴏᴍᴍᴏɴʀᴀᴛɪᴏғɪʀsᴛᴛᴇʀᴍ

ᴛʜᴇ ᴀɴsᴡᴇʀ ᴄᴏᴍᴇs ᴏᴜᴛ ᴛᴏ ʙᴇ 1083