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Question

D and E are points on the sides AB and AC respectively of a ABC. In each of the following cases, determine whether DEBC or not.


(i) AD=5.7 cm, DB=9.5 cm, BD=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.

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Solution

(i) We have:

ADDB = 5.79.5 = 0.6 cmAEEC= 4.88 = 0.6 cmHence,ADDB=AEECApplying the converse of Thales' theorem, we conclude that DEBC.

(ii)
We have:
AB = 11.7 cm, DB = 6.5 cm
Therefore,
AD = 11.7 - 6.5 = 5.2 cm
Similarly,
AC = 11.2 cm, AE = 4.2 cm
Therefore,
EC = 11.2 - 4.2 = 7 cm

Now,ADDB = 5.26.5=45AEEC = 4.27Thus, ADDBAEEC

Applying the converse of Thales' theorem,
we conclude that DE is not parallel to BC.

(iii)
We have:
AB = 10.8 cm, AD = 6.3 cm
Therefore,
DB = 10.8 - 6.3 = 4.5 cm
Similarly,
AC = 9.6 cm, EC = 4 cm
Therefore,
AE = 9.6 - 4 = 5.6 cm
Now,
ADDB=6.34.5=75AEEC=5.64=75ADDB=AEECApplying the converse of Thales' theorem, we conclude that DEBC.

(iv)
We have:
AD = 7.2 cm, AB = 12 cm
Therefore,
DB = 12 - 7.2 = 4.8 cm
Similarly,
AE = 6.4 cm, AC = 10 cm
Therefore,
EC = 10 - 6.4 = 3.6 cm
Now,
ADDB = 7.24.8=32AEEC = 6.43.6= 169Thus, ADDBAEECApplying the converse of Thales' theorem, we conclude that DE is not parallel to BC.

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