Question

Fig. 6.32
6. In Fig. 6.33. PQ and RS are two mirrors placed
parallel to each other. An incident ray AB strikes
the mirror PQ at B. the reflected ray moves along
the path BC and strikes the mirror RS at C and
again reflects back along CD. Prove that
ABCD
Fig. 6.33​

Answer 1

$\huge \bf \red{ \underline{ \underline \mathcal{question : - }}}$

ɪɴ ғɪɢ. 6.33. ᴘǫ ᴀɴᴅ ʀs ᴀʀᴇ ᴛᴡᴏ ᴍɪʀʀᴏʀs ᴘʟᴀᴄᴇᴅ

ᴘᴀʀᴀʟʟᴇʟ ᴛᴏ ᴇᴀᴄʜ ᴏᴛʜᴇʀ. ᴀɴ ɪɴᴄɪᴅᴇɴᴛ ʀᴀʏ ᴀʙ sᴛʀɪᴋᴇs

ᴛʜᴇ ᴍɪʀʀᴏʀ ᴘǫ ᴀᴛ ʙ. ᴛʜᴇ ʀᴇғʟᴇᴄᴛᴇᴅ ʀᴀʏ ᴍᴏᴠᴇs ᴀʟᴏɴɢ

ᴛʜᴇ ᴘᴀᴛʜ ʙᴄ ᴀɴᴅ sᴛʀɪᴋᴇs ᴛʜᴇ ᴍɪʀʀᴏʀ ʀs ᴀᴛ ᴄ ᴀɴᴅ

ᴀɢᴀɪɴ ʀᴇғʟᴇᴄᴛs ʙᴀᴄᴋ ᴀʟᴏɴɢ ᴄᴅ. ᴘʀᴏᴠᴇ ᴛʜᴀᴛ

ᴀʙ||ᴄᴅ

$\huge \bf \red{ \underline{ \underline \mathcal{answer : - }}}$

$\mathbb{let \: draw : - }$

ʙᴍ ⊥ ᴘǫ ᴀɴᴅ ᴄɴ ⊥ ʀs.

$\mathbb{given \: that : - }$

ᴘǫ || ʀs sᴏ ᴛʜᴀᴛ ʙᴍ || ᴄɴ

ᴜsᴇ ᴛʜᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴀʟᴛᴇʀɴᴀᴛᴇ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇs:-

∠2 = ∠3 (1)

∠ᴀʙᴄ = ∠1 + ∠2

ʙᴜᴛ ∠1 = ∠2 sᴏ ᴛʜᴀᴛ

∠ᴀʙᴄ = ∠2 + ∠2

∠ᴀʙᴄ = 2∠2

sɪᴍɪʟᴀʀʟʏ

∠ʙᴄᴅ = ∠3 + ∠4

ʙᴜᴛ ∠3 = ∠4 sᴏ ᴛʜᴀᴛ

∠ ʙᴄᴅ = ∠3 + ∠3

∠ ʙᴄᴅ = 2∠3

ғʀᴏᴍ ᴇǫᴜᴀᴛɪᴏɴ ғɪʀsᴛ

∠ᴀʙᴄ = ∠ᴅᴄʙ

ᴛʜᴇsᴇ ᴀʀᴇ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇs sᴏ ᴛʜᴀᴛ ᴀʙ || ᴄᴅ

ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ

Answer 2

$\huge \purple{ \mid{ \underline{ \overline{✴卂ɴsᴡᴇʀ✴}} \mid}}$

ʙᴍ ⊥ ᴘǫ ᴀɴᴅ ᴄɴ ⊥ ʀs.

ᴘǫ || ʀs sᴏ ᴛʜᴀᴛ ʙᴍ || ᴄɴ

ᴜsᴇ ᴛʜᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴀʟᴛᴇʀɴᴀᴛᴇ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇs:-

∠2 = ∠3--------- (1)

∠ᴀʙᴄ = ∠1 + ∠2

ʙᴜᴛ ∠1 = ∠2 sᴏ ᴛʜᴀᴛ

∠ᴀʙᴄ = ∠2 + ∠2

∠ᴀʙᴄ = 2∠2

sɪᴍɪʟᴀʀʟʏ

∠ʙᴄᴅ = ∠3 + ∠4

ʙᴜᴛ ∠3 = ∠4 sᴏ ᴛʜᴀᴛ

∠ ʙᴄᴅ = ∠3 + ∠3

∠ ʙᴄᴅ = 2∠3

ғʀᴏᴍ ᴇǫᴜᴀᴛɪᴏɴ ғɪʀsᴛ

∠ᴀʙᴄ = ∠ᴅᴄʙ

ᴛʜᴇsᴇ ᴀʀᴇ ᴀʟᴛᴇʀɴᴀᴛᴇ ᴀɴɢʟᴇs sᴏ ᴛʜᴀᴛ ᴀʙ || ᴄᴅ

ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ

hope it's helpful ☺️

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