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Q. ​In triangle PQR, MN||QR and  P M M Q = 2 3

(i) Find  M N Q R
(ii) Prove that triangle OMN and ORQ are similar
(iiI) Find, area of triangle OMN : area of triangle ORQ

Dear Student ,
 
Please find below the solution to the asked query :

PM over MQ equals 2 over 3 MQ over PM equals 3 over 2 MQ over PM plus 1 equals 3 over 2 plus 1 fraction numerator MQ space plus space PM over denominator PM end fraction space equals fraction numerator 3 plus 2 over denominator 2 end fraction PQ over PM equals 5 over 2 In space triangle PMN space and space triangle PQR angle PMN space equals space angle PQR space space space space space space space open parentheses space Corresponding space angle close parentheses angle PNM space equals space angle PRQ space space space space space space space open parentheses space Corresponding space angle close parentheses By space AA space similarity space criterion comma triangle PMN space tilde space triangle PQR So comma open parentheses straight i close parentheses space PQ over PM equals QR over MN 5 over 2 equals QR over MN bold MN over bold QR bold equals bold 2 over bold 5 open parentheses II close parentheses space In space triangle OMN space and space triangle ORQ angle OMN equals angle ORQ space space space space space space space space space space space space space space space space space space space space space space open parentheses space Alternate space interior space angles space close parentheses angle ONM space equals space angle OQR space space space space space space space space space space space space space space space space space space space space open parentheses space Alternate space interior space angles space close parentheses By space AA space similarity space criterion bold triangle bold OMN bold space bold tilde bold space bold triangle bold ORQ open parentheses iii close parentheses space fraction numerator ar space open parentheses space triangle OMN close parentheses over denominator ar space open parentheses space triangle ORQ close parentheses end fraction equals open parentheses MN over RQ close parentheses squared equals open parentheses 2 over 5 close parentheses squared equals 4 over 25 space space space space ANS...
 
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