Question

Q. In the fig,
PS = 3, SQ = 6, PT = x and TR = y. Give any two pairs of values of x and y such that line ST || side QR.

Thanks! :) ​

Answer 1

$\huge\fcolorbox{black}{aqua}{Solution:-}$

✏️ ǫᴜᴇsᴛɪᴏɴ :-

★ ɪɴ ᴛʜᴇ ɢɪᴠᴇɴ ғɪɢᴜʀᴇ, ᴘs = 3, sǫ = 6, ǫʀ = 5, ᴘᴛ = x ᴀɴᴅ ᴛʀ = ʏ. ᴘʀᴏᴠɪᴅᴇ ᴀɴʏ ᴛᴡᴏ ᴘᴀɪʀs ᴏғ ᴠᴀʟᴜᴇ ғᴏʀ x ᴀɴᴅ ʏ. ɢɪᴠᴇ ᴀɴʏ ᴛᴡᴏ ᴘᴀɪʀs ᴏғ ᴠᴀʟᴜᴇ ᴏғ x ᴀɴᴅ ʏ sᴜᴄʜ ᴛʜᴀᴛ ʟɪɴᴇ sᴛ || ǫʀ.

✏️ ᴀɴsᴡᴇʀ :-

★ ᴛʜᴇ ʀᴇǫᴜɪʀᴇᴅ ᴘᴀɪʀs ᴏғ ᴠᴀʟᴜᴇs ᴏғ x ᴀɴᴅ ʏ ᴀʀᴇ 3, 4 ᴀɴᴅ 6, 8 ʀᴇsᴘᴇᴄᴛɪᴠᴇʟʏ.

✏️ sᴛᴇᴘ-ʙʏ-sᴛᴇᴘ ᴇxᴘʟᴀɴᴀᴛɪᴏɴ :-

★ ɢɪᴠᴇɴ :-

• ᴘs = 3;

• sǫ = 6;

• ǫʀ = 5;

• ᴘᴛ = x ᴀɴᴅ,

• ᴛʀ = ʏ

★ ᴛᴏ ᴘʀᴏᴠɪᴅᴇ :-

• ᴛʜᴇ ᴠᴀʟᴜᴇs ᴏғ x ᴀɴᴅ ʏ ᴡʜɪᴄʜ ᴡɪʟʟ sᴀᴛɪsғʏ ᴛʜᴇ ғᴏʟʟᴏᴡɪɴɢ ᴄᴏɴᴅɪᴛɪᴏɴ, ɪ.ᴇ., sᴛ || ǫʀ.

★ ʀᴀᴛɪᴏɴᴀʟᴇ :-

∵ sᴛ || ǫʀ,

ᴛʜᴜs, ɪɴ ᴀᴄᴄᴏʀᴅᴀɴᴄᴇ ᴡɪᴛʜ ᴛʜᴇ ᴛʜᴀʟᴇs ᴛʜᴇᴏʀᴇᴍ [ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ],

ᴘᴛ/ᴘʀ = ᴘs/ᴘǫ

ᴛʜᴇʀᴇғᴏʀᴇ,

ᴏɴ ᴘʟᴜɢɢɪɴɢ ɪɴ ᴛʜᴇ ʀᴇǫᴜɪʀᴇᴅ ᴠᴀʟᴜᴇs, ᴡᴇ ᴀᴄǫᴜɪʀᴇ,

x/x + ʏ = 3/9

⇒x/x + ʏ = 1/3

⇒3x = x + ʏ

⇒2x = ʏ

ɴᴏᴡ,

ғʀᴏᴍ ᴛʜᴇ ɢɪᴠᴇɴ sᴄᴇɴᴀʀɪᴏ, ᴀssᴜᴍɪɴɢ ᴛᴡᴏ ᴘᴀɪʀs ᴏғ sᴀᴛɪsғʏɪɴɢ ᴠᴀʟᴜᴇs ғᴏʀ ᴛʜᴇ ᴄᴏɴᴅɪᴛɪᴏɴ, ᴡᴇ ɢᴇᴛ,

• x = 3 ᴀɴᴅ ʏ = 6

&

• x = 4 ᴀɴᴅ sᴀʏ ʏ = 8

[ɴᴏᴛᴇ :- ᴀɴʏ ᴠᴀʟᴜᴇs ᴏғ x ᴀɴᴅ ʏ ᴄᴀɴ ʙᴇ ɪɴsᴇʀᴛᴇᴅ ᴜɴʟᴇss ɪᴛ sᴀᴛɪsғɪᴇs ᴛʜᴇ ʀᴇʟᴀᴛɪᴏɴ, 2x = ʏ]

▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃▃

❏ ᴀᴅᴅɪᴛɪᴏɴᴀʟ ɪɴғᴏʀᴍᴀᴛɪᴏɴ :-

★ ᴡʜᴏ ᴡᴀs ᴛʜᴀʟᴇs ?!

• ᴛʜᴀʟᴇs ᴏғ ᴍɪʟᴇᴛᴜs ᴡᴀs ᴀ ɢʀᴇᴇᴋ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄɪᴀɴ, ᴀsᴛʀᴏɴᴏᴍᴇʀ ᴀɴᴅ ᴘʀᴇ-sᴏᴄʀᴀᴛɪᴄ ᴘʜɪʟᴏsᴏᴘʜᴇʀ ғʀᴏᴍ ᴍɪʟᴇᴛᴜs ɪɴ ɪᴏɴɪᴀ, ᴀsɪᴀ ᴍɪɴᴏʀ. ʜᴇ ᴡᴀs ᴏɴᴇ ᴏғ ᴛʜᴇ sᴇᴠᴇɴ sᴀɢᴇs ᴏғ ɢʀᴇᴇᴄᴇ. ʜᴇ ᴄᴏɪɴᴇᴅ ᴀɴ ɪᴍᴘᴏʀᴛᴀɴᴛ ᴛʀᴜᴛʜ ʀᴇʟᴀᴛɪɴɢ ᴛᴏ ᴛᴡᴏ ᴇǫᴜɪᴀɴɢᴜʟᴀʀ ᴛʀɪᴀɴɢʟᴇs ᴡʜɪᴄʜ ɪs ᴀs ғᴏʟʟᴏᴡs :-

"ᴛʜᴇ ʀᴀᴛɪᴏ ᴏғ ᴀɴʏ ᴛᴡᴏ ᴄᴏʀʀᴇsᴘᴏɴᴅɪɴɢ sɪᴅᴇs ɪɴ ᴛᴡᴏ ᴇǫᴜɪᴀɴɢᴜʟᴀʀ ᴛʀɪᴀɴɢʟᴇs ɪs ᴀʟᴡᴀʏs ᴛʜᴇ sᴀᴍᴇ."

ɪᴛ ɪs ʙᴇʟɪᴇᴠᴇᴅ ᴛʜᴀᴛ ʜᴇ ʜᴀᴅ ᴜsᴇᴅ ᴀ ʀᴇsᴜʟᴛ ᴄᴀʟʟᴇᴅ ᴛʜᴇ ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ ( ɴᴏᴡ ᴋɴᴏᴡɴ ᴀs ᴛʜᴀʟᴇs ᴛʜᴇᴏʀᴇᴍ ) ғᴏʀ ᴛʜᴇ sᴀᴍᴇ.

★ ᴡʜᴀᴛ ᴅᴏᴇs ᴛʜᴇ ᴛʜᴀʟᴇs ᴛʜᴇᴏʀᴇᴍ sᴀʏ ?!

• ᴛʜᴇ ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ sᴛᴀᴛᴇs ᴛʜᴀᴛ, ɪғ ᴀ ʟɪɴᴇ 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.

• ᴛʜᴜs, ғᴏʀ ɪɴsᴛᴀɴᴄᴇ, ɪɴ ᴀ ∆ᴀʙᴄ ɪɴ ᴡʜɪᴄʜ ʟɪɴᴇ ᴘᴀʀᴀʟʟᴇʟ ᴛᴏ sɪᴅᴇ ʙᴄ ɪɴᴛᴇʀsᴇᴄᴛs ᴏᴛʜᴇʀ ᴛᴡᴏ sɪᴅᴇs ᴀʙ ᴀɴᴅ ᴀᴄ ᴀɴᴅ ᴇ ʀᴇsᴘᴇᴄᴛɪᴠᴇʟʏ, ᴛʜᴇɴ,

ᴀᴅ/ᴅʙ = ᴀᴇ/ᴇᴄ

• ᴏᴛʜᴇʀ ʀᴇʟᴀᴛɪᴏɴs ɪɴᴄʟᴜᴅᴇ —

☞ ᴀᴅ/ᴀʙ = ᴀᴇ/ᴀᴄ

☞ ᴅʙ/ᴀʙ = ᴇᴄ/ᴀᴄ

☞ ᴀʙ/ᴀᴅ = ᴀᴄ/ᴀᴇ

☞ᴀʙ/ᴅʙ = ᴀᴄ/ᴇᴄ

❏ ʀᴇʟᴀᴛᴇᴅ :-

★ sɪᴍɪʟᴀʀɪᴛʏ ᴏғ ᴛʀɪᴀɴɢʟᴇs –

• ᴛᴡᴏ ᴛʀɪᴀɴɢʟᴇs ᴀʀᴇ sᴀɪᴅ ᴛᴏ ʙᴇ sɪᴍɪʟᴀʀ, ɪғ

☞ ᴛʜᴇɪʀ ᴄᴏʀʀᴇsᴘᴏɴᴅɪɴɢ ᴀɴɢʟᴇs ᴀʀᴇ ᴇǫᴜᴀʟ ᴀɴᴅ,

☞ ᴛʜᴇɪʀ ᴄᴏʀʀᴇsᴘᴏɴᴅɪɴɢ sɪᴅᴇs ᴀʀᴇ ɪɴ ᴛʜᴇ sᴀᴍᴇ ʀᴀᴛɪᴏ ( ᴏʀ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ ).

• ɴᴏᴛᴇ - ɪғ ᴄᴏʀʀᴇsᴘᴏɴᴅɪɴɢ ᴀɴɢʟᴇs ᴏғ ᴛᴡᴏ ᴛʀɪᴀɴɢʟᴇs ᴀʀᴇ ᴇǫᴜᴀʟ, ᴛʜᴇɴ ᴛʜᴇʏ ᴀʀᴇ ᴛᴇʀᴍᴇᴅ ᴀs ᴇǫᴜɪᴀɴɢᴜʟᴀʀ ᴛʀɪᴀɴɢʟᴇs.

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í հօթҽ Եհís հҽlթs! :)