Question

In parallelogram $ABCD$, $E$ is mid-point of side $AB$ and $CE$ meets the diagonal $BD$ at point $O$, then $OE:OC$ is,

• A. $5:2$
• B. $1:2$
• C. $9:2$
• D. $7:2$

Answer 1

The correct option is B $1:2$

Given, Parallelogram $ABCD$ with $E$ as a mid point of $AB,CE$ meets diagonal $BD$ at $O$.

Now, In $△EOB$ and $△DOC$

$\angle DOC=\angle EOB$ (Vertically opposite angles are equal.)

$\angle EBO=\angle ODC$ (Alternate angles, $\because AB\parallel CD$ and $BD$ intersect it)

$\angle OCD=\angle OEB$ (Alternate angles, $\because AB\parallel CD$ and $EC$ intersect it)

Using $A-A-A$ property, $△EOB\sim △DOC$
Using similarity property,

$\frac{OE}{OC}=\frac{EB}{DC}$

$\frac{OE}{OC}=\frac{EB}{AB}$

(As $AB=DC$,opposite sides of parallelogram)
$AB=2EB$

(As $E$ is mid point of AB)
$\therefore \frac{OE}{OC}=\frac{1}{2}$

So, $OE:OC=1:2$