In the given figure DE || BC and AE : EC = 4 : 7, then BC: DE = ……

1.	In the given figure DE \|\| BC and AE : EC = 4 : 7, then BC: DE = …………(A) 11 : 4 (B) 4 : 11 (C) 11 : 3 (D) 3 : 11
Answer» Correct option is (A) 11 : 4 We have DE \|\| BC and \(\frac{AE}{EC}=\frac47\) \(\Rightarrow\) \(EC=\frac74AE\) \(\because AC=AE+EC\) \(=AE+\frac74AE\) \(=\frac{11}4AE\) \(\frac{AC}{AE}=\frac{11}4\) ________________(1) In triangles \(\triangle ABC\;\&\;\triangle ADE,\) \(\angle ABC=\angle ADE,\) \(\angle ACB=\angle AED\) (Corresponding angles as DE \|\| BC) \(\angle BAC=\angle DAE\) (Common angle) \(\therefore\) \(\triangle ABC\sim\triangle ADE\) (By AAA similarity rule) \(\therefore\) \(\frac{BC}{DE}=\frac{AC}{AE}=\frac{11}4\) (From (1)) \(\therefore\) BC : DE = 11 : 4 Correct option is: (A) 11 : 4

In the given figure DE || BC and AE : EC = 4 : 7, then BC: DE = …………(A) 11 : 4 (B) 4 : 11 (C) 11 : 3 (D) 3 : 11

Answer»

Correct option is (A) 11 : 4

We have DE || BC

and \(\frac{AE}{EC}=\frac47\)

\(\Rightarrow\) \(EC=\frac74AE\)

\(\because AC=AE+EC\)

\(=AE+\frac74AE\)

\(=\frac{11}4AE\)

\(\frac{AC}{AE}=\frac{11}4\) ________________(1)

In triangles \(\triangle ABC\;\&\;\triangle ADE,\)

\(\angle ABC=\angle ADE,\)

\(\angle ACB=\angle AED\) (Corresponding angles as DE || BC)

\(\angle BAC=\angle DAE\) (Common angle)

\(\therefore\) \(\triangle ABC\sim\triangle ADE\) (By AAA similarity rule)

\(\therefore\) \(\frac{BC}{DE}=\frac{AC}{AE}=\frac{11}4\) (From (1))

\(\therefore\) BC : DE = 11 : 4

Correct option is: (A) 11 : 4