From the given figure, ar (ΔADE): ar (ΔABC) =(A) 25 : 9 (B) 9 : 64�

1.	From the given figure, ar (ΔADE): ar (ΔABC) =(A) 25 : 9 (B) 9 : 64 (C) 25 : 64 (D) 9 : 25
Answer» Correct option is (B) 9 : 64 \(\because DE\bot AC\;\&\;BC\bot AC\) \(\therefore DE\,\|\|\,BC\) Now in triangles \(\triangle ADE\;\&\;\triangle ABC,\) \(\angle ADE=\angle ABC,\) (Corresponding angles as DE \|\| BC) \(\angle AED=\angle ACB\) \(=90^\circ\) \(\angle DAE=\angle BAC\) (Common angle) \(\therefore\) \(\triangle ADE\sim\triangle ABC\) (By AAA similarity rule) \(\therefore\frac{ar(\triangle ADE)}{ar(\triangle ABC)}=(\frac{AD}{AB})^2\) \(=(\frac{DE}{BC})^2=(\frac{AE}{EC})^2\) \(\Rightarrow\frac{ar(\triangle ADE)}{ar(\triangle ABC)}=(\frac{AE}{AE+EC})^2\) \(=(\frac{3}{3+5})^2=(\frac{3}{8})^2\) \(=\frac{3^2}{8^2}=\frac9{64}\) \(\therefore\) \(ar(\triangle ADE):ar(\triangle ABC)\) = 9:64 Correct option is: (B) 9 : 64

From the given figure, ar (ΔADE): ar (ΔABC) =(A) 25 : 9 (B) 9 : 64 (C) 25 : 64 (D) 9 : 25

Answer»

Correct option is (B) 9 : 64

\(\because DE\bot AC\;\&\;BC\bot AC\)

\(\therefore DE\,||\,BC\)

Now in triangles \(\triangle ADE\;\&\;\triangle ABC,\)

\(\angle ADE=\angle ABC,\) (Corresponding angles as DE || BC)

\(\angle AED=\angle ACB\) \(=90^\circ\)

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

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

\(\therefore\frac{ar(\triangle ADE)}{ar(\triangle ABC)}=(\frac{AD}{AB})^2\)

\(=(\frac{DE}{BC})^2=(\frac{AE}{EC})^2\)

\(\Rightarrow\frac{ar(\triangle ADE)}{ar(\triangle ABC)}=(\frac{AE}{AE+EC})^2\)

\(=(\frac{3}{3+5})^2=(\frac{3}{8})^2\)

\(=\frac{3^2}{8^2}=\frac9{64}\)

\(\therefore\) \(ar(\triangle ADE):ar(\triangle ABC)\) = 9:64

Correct option is: (B) 9 : 64