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Question

In a ΔABC, D and E are points on the sides AB and AC respectively. For each of the following cases show that DE || BC: 
(i) AB = 12cm, AD = 8 cm, AE = 12 cm and AC= 18cm. 
(ii)AB = 5.6cm, AD= 1.4cm, AC = 7.2 cm and AE= 1.8 cm. 
(iii) AB= 10.8 cm, BD= 4.5cm, AC = 4.8cm and AE =2.8 cm. 
(iv)AD = 5.7cm, BD = 9.5cm, AE = 3.3cm and EC = 5.5 cm.


Solution

DE will be || to BC when fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction  ( it shows that ABC and ADE are similar triangles with common angle A, Hence the other two corresponding angles will be equal too)

(i) fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction
   8 over 12 equals 12 over 18 2 over 3 equals 2 over 3
Hence DE|| BC

(ii)fraction numerator A D over denominator A B space end fraction equals fraction numerator 1.4 over denominator 5.6 end fraction equals 1 fourth
      fraction numerator A E over denominator A C space end fraction equals fraction numerator 1.8 over denominator 7.2 end fraction equals 1 fourth
       fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction
Hence  DE|| BC

(iii) fraction numerator A D over denominator A B space end fraction equals fraction numerator 10.8 minus 4.5 over denominator 10.8 end fraction equals fraction numerator 6.3 over denominator 10.8 end fraction equals 7 over 12
      fraction numerator A E over denominator A C space end fraction equals fraction numerator 2.8 over denominator 4.8 end fraction equals 7 over 12
       fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction
Hence  DE|| BC

(iv)fraction numerator A D over denominator A B space end fraction equals fraction numerator 5.7 over denominator 5.7 plus 9.5 end fraction equals fraction numerator 5.7 over denominator 15.2 end fraction equals 3 over 8
      fraction numerator A E over denominator A C space end fraction equals fraction numerator 3.3 over denominator 3.3 plus 5.5 end fraction equals 3 over 8
       fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction
Hence  DE|| BC


Mathematics
RD Sharma
Standard X

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