Question

ABCD is a trapezium in which AB||DC and its diagonals intersect each other at point o. Show that: AO/BO= CO/DO

Draw the diagram of ur own-​​

Answer 1

*＊✿❀ ʜᴇʟʟᴏ ❀✿＊*

ʏᴏᴜʀ ᴀɴsᴡᴇʀ =>

ɢɪᴠᴇɴ ᴘᴀʀᴀᴍᴇᴛᴇʀs

ᴀʙᴄᴅ ɪs ᴀ ᴛʀᴀᴘᴇᴢɪᴜᴍ ᴡʜᴇʀᴇ ᴀʙ || ᴅᴄ ᴀɴᴅ ᴅɪᴀɢᴏɴᴀʟs ᴀᴄ ᴀɴᴅ ʙᴅ ɪɴᴛᴇʀsᴇᴄᴛ ᴀᴛ ᴏ.

ᴛᴏ ᴘʀᴏᴠᴇ

ᴀᴏ/ʙᴏ=ᴄᴏ/ᴅᴏ

ᴄᴏɴsᴛʀᴜᴄᴛɪᴏɴ

ᴅʀᴀᴡ ᴀ ʟɪɴᴇ ᴇғ ᴘᴀssɪɴɢ ᴛʜʀᴏᴜɢʜ ᴏ ᴀɴᴅ ᴀʟsᴏ ᴘᴀʀᴀʟʟᴇʟ ᴛᴏ ᴀʙ

ɴᴏᴡ, ᴀʙ ll ᴄᴅ

ʙʏ ᴄᴏɴsᴛʀᴜᴄᴛɪᴏɴ ᴇғ ll ᴀʙ

∴ ᴇғ ll ᴄᴅ

ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ Δᴀᴅᴄ,

ᴡʜᴇʀᴇ ᴇᴏ ll ᴀʙ

ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ

ᴀᴇ/ᴇᴅ=ᴀɪ/ᴏᴄ ………………………………(1)

ɴᴏᴡ ᴄᴏɴsɪᴅᴇʀ Δ ᴀʙᴅ

ᴡʜᴇʀᴇ ᴇᴘ ll ᴀʙ

ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ʙᴀsɪᴄ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ ᴛʜᴇᴏʀᴇᴍ

ᴀᴇ/ᴇᴅ=ʙᴏ/ᴏᴅ ……………………………..(2)

ғʀᴏᴍ ᴇǫᴜᴀᴛɪᴏɴ (1) ᴀɴᴅ (2) ᴡᴇ ʜᴀᴠᴇ

ᴀᴏ/ᴏᴄ=ʙᴏ/ᴏᴅ

⇒ ᴀᴏ/ʙᴏ=ᴏᴄ/ᴏᴅ

ʜᴇɴᴄᴇ ᴛʜᴇ ᴘʀᴏᴏғ.

Answer 2

Step-by-step explanation:

Given parameters

ABCD is a trapezium where AB || DC and diagonals AC and BD intersect at O.

To prove

AOBO=CODO

Construction

Draw a line EF passing through O and also parallel to AB

Now, AB ll CD

By construction EF ll AB

∴ EF ll CD

Consider the ΔADC,

Where EO ll AB

According to basic proportionality theorem

AEED=AOOC ………………………………(1)

Now consider Δ ABD

where EO ll AB

According to basic proportionality theorem

AEED=BOOD ……………………………..(2)

From equation (1) and (2) we have

AOOC=BOOD

⇒ AOBO=OCOD

Hence the proof.