Question

In the figure shown above, AC || BD and CE || DF. If OA = 12 cm, AB = 9 cm, OC = 8 cm and EF = 4.5 cm, find OE.

Answer 1

Given that, in $\Delta OBD,AC||BD,OA=12cm,AB=9cm,andOC=8cm$.

To find out: The length of $OE$.

According to the basic proportionality theorem, if a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points, then it divides the two sides in the same ratio.

Hence, using the basic proportionality theorem in $\Delta OBD$, we have:

$\frac{OA}{BA}=\frac{OC}{DC}$

$\Rightarrow \frac{12}{9}=\frac{8}{DC}$

$\Rightarrow 12DC=8\times 9$

$\Rightarrow 12DC=72$

$\Rightarrow DC=\frac{72}{12}$

$\therefore DC=6cm$

Now, in $\Delta ODF,CE||DF,EF=4.5cm,DC=6cmandOC=8cm$.

Using the basic proportionality theorem in $\Delta ODF$, we have

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

$\Rightarrow \frac{8}{6}=\frac{OE}{4.5}$

$\Rightarrow 6OE=8\times 4.5$

$\Rightarrow 6OE=36$

$\Rightarrow OE=\frac{36}{6}$

$\therefore OE=6$

Hence, the required length of $OE$ is $6cm$.