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Question

D and E are points on the sides AB and AC respectively of a ΔABC. In each of the follwing cases, determine whether DE || BC or not.

(i) AD = 5.7 cm, DB = 9.5 cm, AE = 4.8 cm and EC = 8 cm.

(ii) AB = 11.7 cm, AC = 11.2 cm, BD = 6.5 cm and AE = 4.2 cm.

(iii) AB = 10.8 cm, AD = 6.3 cm, AC = 9.6 cm and EC = 4 cm.

(iv) AD = 7.2 cm, AE = 6.4 cm, AB = 12 cm and AC = 10 cm.


Solution

DE || BCif ADE and ABC are similar

(i) ADDB= 5.79.5=0.6

AEEC= 4.88=0.6

therefore DE || BC

i n space increment A B C space a n d space increment A D E space a r e space si m i l i a r space t h e n D E parallel to B C 2 right parenthesis a s s u m e space increment A B C space a n d space increment A D E space a r e space si m i l i a r t h e n fraction numerator A B over denominator A D end fraction equals fraction numerator A C over denominator A E end fraction A D equals A B minus B D equals 11.7 minus 6.5 equals 5.2 fraction numerator 11.7 over denominator 5.2 end fraction equals fraction numerator 11.2 over denominator 4.2 end fraction 2.25 not equal to 2.67 i n space t h i s space c a s e space D E not parallel to B C 3 right parenthesis a s s u m e space increment A B C space a n d space increment A D E space a r e space si m i l i a r t h e n fraction numerator A B over denominator A D end fraction equals fraction numerator A C over denominator A E end fraction A E equals A C minus E C equals 9.6 minus 4 equals 5.6 fraction numerator 10.8 over denominator 6.3 end fraction equals fraction numerator 9.6 over denominator 5.6 end fraction 1.714 equals 1.714 i n space t h i s space c a s e D E parallel to B C 4 right parenthesis a s s u m e space increment A B C space a n d space increment A D E space a r e space si m i l i a r t h e n fraction numerator A B over denominator A D end fraction equals fraction numerator A C over denominator A E end fraction fraction numerator 12 over denominator 7.2 end fraction equals fraction numerator 10 over denominator 6.4 end fraction 1.67 not equal to 1.56 i n space t h i s space c a s e i n space t h i s space c a s e space D E not parallel to B C

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