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ICSE Class 10 Answered

In triangle PQR,ST is drawn parallel to QR such that PT:RT=2:3.if O is the point of intersection of QT and RS,then ratio of the areas of triangle OTS and triangle OQR will be.
Asked by nitinkumarsingh.singh.75 | 16 Oct, 2015, 01:29: PM
answered-by-expert Expert Answer
C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
G i v e n space t h a t space S T space i s space d r a w n space p a r a l l e l space t o space Q R space s u c h space t h a t space P T colon R T equals 2 colon 3 T h u s angle P S T equals angle P Q R a n d angle P T S equals angle P R Q T h e r e f o r e comma space b y space A n g l e minus A n g l e space s i m i l a r i t y comma space triangle P S T tilde triangle P Q R B y space p r o p o r t i o n a l i t y space t h e o r e m comma space w e space h a v e comma fraction numerator P T over denominator R T end fraction equals fraction numerator P S over denominator S Q end fraction equals 2 over 3 rightwards double arrow fraction numerator R T over denominator P T end fraction equals 3 over 2 rightwards double arrow 1 plus fraction numerator R T over denominator P T end fraction equals 1 plus 3 over 2 rightwards double arrow fraction numerator P T plus R T over denominator P T end fraction equals fraction numerator 2 plus 3 over denominator 2 end fraction rightwards double arrow fraction numerator P R over denominator P T end fraction equals 5 over 2 rightwards double arrow fraction numerator P T over denominator P R end fraction equals 2 over 5 rightwards double arrow fraction numerator P S over denominator P Q end fraction equals fraction numerator S T over denominator Q R end fraction equals fraction numerator P T over denominator P R end fraction equals 2 over 5... left parenthesis 1 right parenthesis C o n s i d e r space t h e space t r i a n g l e s space O S T space a n d space O Q R. angle S O T equals angle Q O R space space space space open square brackets because v e r t i c a l l y space o p p o s i t e space a n g l e s close square brackets angle O S T equals angle O R Q space space space space space space open square brackets because S T parallel to Q R close square brackets T h u s comma space triangle O S T tilde triangle O R Q T h e space r a t i o space o f space t h e space a r e a s space o f space s i m i l a r space t r i a n g l e s space i s space e q u a l space t o space t h e space r a t i o space o f space t h e i r space c o r r e s p o n d i n g space s i d e s. S T space a n d space Q R space a r e space t h e space c o r r e s p n d i n g space s i d e s space o f space t h e space t r i a n g l e s comma space triangle O S T space a n d triangle O R Q T h u s comma space fraction numerator A r e a open parentheses O S T close parentheses over denominator A r e a open parentheses O R Q close parentheses end fraction equals fraction numerator S T squared over denominator Q R squared end fraction rightwards double arrow fraction numerator A r e a open parentheses O S T close parentheses over denominator A r e a open parentheses O R Q close parentheses end fraction equals open parentheses fraction numerator S T over denominator Q R end fraction close parentheses squared rightwards double arrow fraction numerator A r e a open parentheses O S T close parentheses over denominator A r e a open parentheses O R Q close parentheses end fraction equals open parentheses 2 over 5 close parentheses squared space space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets rightwards double arrow fraction numerator A r e a open parentheses O S T close parentheses over denominator A r e a open parentheses O R Q close parentheses end fraction equals 2 squared over 5 squared rightwards double arrow fraction numerator A r e a open parentheses O S T close parentheses over denominator A r e a open parentheses O R Q close parentheses end fraction equals 4 over 25
Answered by Vimala Ramamurthy | 17 Oct, 2015, 09:35: AM

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