The diagonals of a trapezium ABCD with AB || DC, intersect each other

1.	The diagonals of a trapezium ABCD with AB \|\| DC, intersect each other at the point O. If AB = 2 CD, then the ratio of areas of triangles AOB and COD is(A) 4 : 1 (B) 1 : 4 (C) 3 : 4 (D) 4 : 3
Answer» Correct option is (A) 4 : 1 In triangles \(\triangle AOB\;and\;\triangle COD,\) \(\angle OAB=\angle OCD\) (Alternative interior angles as AB \|\| DC) \(\angle OBA=\angle ODC\) (Alternative interior angles as AB \|\| DC) \(\angle AOB=\angle COD\) (Vertically opposite angles) \(\therefore\) \(\triangle AOB\sim\triangle COD\) (By AAA similarity rule) \(\therefore\) \(\frac{ar(\triangle AOB)}{ar(\triangle COD)}=(\frac{AB}{CD})^2\) (By properties of similar triangle) \(=(\frac21)^2=\frac41\) \((\because AB=2CD\Rightarrow\frac{AB}{CD}=\frac21)\) \(\therefore\) \(ar(\triangle AOB):ar(\triangle COD)=4:1\) Correct option is: (A) 4 : 1

The diagonals of a trapezium ABCD with AB || DC, intersect each other at the point O. If AB = 2 CD, then the ratio of areas of triangles AOB and COD is(A) 4 : 1 (B) 1 : 4 (C) 3 : 4 (D) 4 : 3

Answer»

Correct option is (A) 4 : 1

In triangles \(\triangle AOB\;and\;\triangle COD,\)

\(\angle OAB=\angle OCD\) (Alternative interior angles as AB || DC)

\(\angle OBA=\angle ODC\) (Alternative interior angles as AB || DC)

\(\angle AOB=\angle COD\) (Vertically opposite angles)

\(\therefore\) \(\triangle AOB\sim\triangle COD\) (By AAA similarity rule)

\(\therefore\) \(\frac{ar(\triangle AOB)}{ar(\triangle COD)}=(\frac{AB}{CD})^2\) (By properties of similar triangle)

\(=(\frac21)^2=\frac41\) \((\because AB=2CD\Rightarrow\frac{AB}{CD}=\frac21)\)

\(\therefore\) \(ar(\triangle AOB):ar(\triangle COD)=4:1\)

Correct option is: (A) 4 : 1