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Question

In Fig. 7.179, DE || BC
(i)If DE= 4 cm, BC = 6cm and Area (ΔADE)=16cm2, find the area of ΔABC
(ii) If DE =4 cm, BC = 8 cm and Area (ΔADE)=25cm2, find the area of ΔABC
(iii) If DE : BC = 3 : 5. Calculate the ratio of the areas of ΔADE and the trapezium BCED.


 


Solution

In ΔADE and ΔABC
∠C = ∠E [DE||BC]
∠A = ∠A [common]
Then, ΔADE ~ ΔABC [By AA similarity]
(i)

fraction numerator a r left parenthesis A B C right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space equals space open parentheses fraction numerator B C over denominator D E end fraction close parentheses squared fraction numerator begin display style a r left parenthesis A B C right parenthesis end style over denominator begin display style 16 end style end fraction space equals open parentheses 6 over 4 close parentheses squared space equals space 36 over 16 h e n c e space a r left parenthesis A B C right parenthesis space equals space 36 space c m squared
(ii)
fraction numerator a r left parenthesis A B C right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space equals space open parentheses fraction numerator B C over denominator D E end fraction close parentheses squared fraction numerator begin display style a r left parenthesis A B C right parenthesis end style over denominator begin display style 25 end style end fraction space equals open parentheses 8 over 4 close parentheses squared space equals space 64 over 16 h e n c e space a r left parenthesis A B C right parenthesis space equals space 100 space c m squared
(iii)
fraction numerator a r left parenthesis A B C right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space equals space open parentheses fraction numerator B C over denominator D E end fraction close parentheses squared fraction numerator a r left parenthesis A B C right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space equals open parentheses 5 over 3 close parentheses squared space equals space 25 over 9 s u b t r a c t i n g space 1 space f r o m space b o t h space s i d e s fraction numerator a r left parenthesis A B C right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space minus 1 space equals space 25 over 9 space minus 1 space fraction numerator a r left parenthesis A B C right parenthesis minus a r left parenthesis A D E right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space equals space 16 over 9 fraction numerator t r a p. left parenthesis B C E D right parenthesis over denominator a r left parenthesis A D E right parenthesis end fraction space equals 16 over 9
 

Mathematics
RD Sharma
Standard X

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