Question

plzz give the correct ans...only if u r sure..u can take your time but ans should b correct​

Answer 1

✪ᴀɴsᴡᴇʀ✪

• $\boxed{\boxed{\blue{ UT=\frac{5PQ}{8}}}}$

★ɢɪᴠᴇɴ★

• A triangle PQR with points U and T on PR and QR such that PU:UR = 3:5 and QT:TR = 3:5.

☆ᴛᴏ ғɪɴᴅ☆

• Value of UT in terms of PQ.

✫ᴄᴏɴsᴛʀᴜᴄᴛɪᴏɴ✫

• Draw a line segment connecting U and T.

⍟ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴ⍟

Now,

$\to \frac{PU}{UR} = \frac{QT}{TR} = \frac{3}{5}$

$\to \frac{PU}{UR} +1 = \frac{QT}{TR} +1= \frac{3}{5}+1$

$\to \frac{PU+UR}{UR} = \frac{QT+TR}{TR}= \frac{3+5}{5}$

$\to \frac{PR}{UR} = \frac{QR}{TR}= \frac{8}{5}...(1)$

In ΔPQR and ΔUTR

$\:\:\:\:\: \frac{PR}{UR} = \frac{QR}{TR}$

$\:\:\:\:\:∠PRQ = ∠URT$( common)

So, by SAS similarity criteria,

$\:\:\:\:\: ΔPQR$ ~ $ΔUTR$

Hence,

$\to \frac{PR}{UR} = \frac{QR}{TR} = \frac{PQ}{UT}$

$\to \frac{PQ}{UT}=\frac{8}{5}$

$\to \frac{UT}{PQ}=\frac{5}{8}$

$\to UT=\frac{5PQ}{8}$