In a ∆ABC, D and E are points on the sides AB and AC respectively. For each of the following cases show that DE || BC :
(i) AB = 12 cm, AD = 8 cm, AE = 12 cm and AC = 18 cm.
(ii) AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 and AE = 1.8 cm.
(iii) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.
(iv) AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm
Answers
Converse of basic proportionality theorem :
If a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
SOLUTION :
1) Given : D and E are the points on sides AB and AC. AB = 12 cm, AD = 8 cm, AE = 12 cm, and AC = 18 cm.
To prove : DE || BC.
DB= AB - AD
DB = 12 - 8
DB = 4 cm
EC = AC - AE
EC = 18 - 12
EC = 6 cm
In ∆ABC,
AD / DB = 8/4 = 2
And, AE/EC = 12/ 6 = 2
so, AD / DB = AE/EC
Hence, DE || BC.
[By Converse of basic proportionality theorem]
2) Given : D and E are the points on sides AB and AC. AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm, and AE = 1.8 cm.
To prove : DE || BC.
DB= AB - AD
DB = 5.6 - 1.4
DB = 4.2 cm
EC = AC - AE
EC = 7.2 - 1.8
EC = 5.4 cm
In ∆ABC,
AD / DB = 1.4 /4.2 = 1/3
And, AE/EC = 1.8/ 5.4 = 1/3
so, AD / DB = AE/EC
Hence, DE || BC.
[By Converse of basic proportionality theorem]
3) Given : D and E are the points on sides AB and AC. AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm, and AE = 2.8 cm
To prove : DE || BC.
AD = AB – DB
AD = 10.8 – 4.5
AD = 6.3 cm
EC = AC – AE
EC = 4.8 – 2.8
EC = 2 cm
In ∆ABC,
AD / DB = 6.3/4.5 = 7/5 = 1.4
And, AE/EC = 2.8 /2.0 = 1.4
so, AD / DB = AE/EC
Hence, DE || BC.
[By Converse of basic proportionality theorem]
4) Given : D and E are the points on sides AB and AC. AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm, and EC = 5.5 cm.
To prove : DE || BC.
In ∆ABC,
AD / DB = 5.7/9.5 = 3/5
And, AE/EC = 3.3/5.5 = 3/5
so, AD / DB = AE/EC
Hence, DE || BC.
[By Converse of basic proportionality theorem]
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